An Application of the Hard Lefschetz Theorem

R.Weissauer

University of Mannheim

December 1993

This note discusses the cohomology groups of locally constant sheaves on Siegel modular threefolds. Notation from [W] will be used freely. Our aim is to show, that irreducible cuspidal automorphic representations, which are not CAP representations in the sense of Piateskii-Shapiro, whose archimedian component belongs to the discret series do only contribute to cohomology in degree 3. The results can be applied to proof the generalized Ramanujan conjecture for such automorphic forms. This includes as a special case holomorphic Siegel modular forms of genus two and weight ≥ 3.

\underline{Notation}: Let Γ be a neat subgroup of finite index in the symplectic group Sp(4, ZZ). Let \overline{X} be a suitable toroidal compactification of the Siegel modular variety X = H/Γ, where H denotes the Siegel upper half space of genus 2. Assume j : X ↪ \overline{X}. Let P = MU be the Siegel parabolic subgroup of upper block triangular matrices. The first upper block is an invertible matrix and this assignment gives isomorphisms M ≅ P/U ≅ Gl(2, Q).

1.Review of Eichler integrals:

To any irreducible algebraic representation λ : GSp(4, Q) → Gl(V_{λ}) of the symplectic group of similitudes GSp(4, Q) is associated a locally constant sheaf E_{λ} on X with fibres V_{λ}. The isomorphim class of this holomorphic vector sheaves depends only on the restriction of the representation λ to the group Sp(4, Q) and all these sheaves are selfdual. The operation of the Hecke algebra and also the Galois action, to be defined later, depends also on the central character of the representation λ of GSp(4, Q). The restricted representation λ is uniquely described by its Sp(4, Q)-dominant weight, which by abuse of notation will be also denoted λ. The direct image complex Rj_{∗}(E_{λ}) on \overline{X} can be resolved using the BGG-complex. This means, that the complex is an quasiisomorphic to a complex \overline{K}_{λ}^{•} ≅ Rj_{∗}(E_{λ}) on \overline{X}, where

\overline{K}_{λ}^{p} = ⨁_{μ} \overline{W} _{μ}^{∗}

and where the maps are certain differential operators. The vectorbundles \overline{W}_{μ}, which occur in this sum are prolongations of vectorbundles W_{μ} on X defined by automorphic cocycles attached to a rational irreducible representation μ of the group Gl(2, Q). By abuse of notation let again μ denote the corresponding Gl(2, Q)-dominant weight. Then the sum extends over all those Gl(2)-dominant weights μ, where μ = w(λ + ρ) − ρ holds for some element w of lenght l(w) = p in the Weyl group. According to Chai-Faltings the spectral sequence

E_{1}^{p,q} = H^{q}(\overline{X}, \overline{K}_{λ}^{p}) ⇒ H^{p+q}(X, E_{λ})

degenerates at the 1-level. See [CF], p.233. In our case this may be made more explicit: The Sp(4) dominant weights may be identified with pairs of integers λ = (λ_{1}, λ_{2}) such that λ_{1} ≥ λ_{2} ≥ 0. Furthermore ρ = (2, 1). There are 4 substitutions in the Weyl group, which map Sp(4)-dominant weights to Gl(2)-dominant ones. These are denoted id, σ, τ, w

where σ(u, v) = (u, −v), τ(u, v) = (v, −u) and w(u, v) = (−v, −u) are of length 1,2 and 3 respectively. Then μ = w(λ + ρ) − ρ is given by (λ_{1}, λ_{2}), (λ_{1}, −λ_{2} − 2), (λ_{2} − 1, −λ_{1} − 3) and

(−λ_{2} − 3, −λ_{1} − 3) respectively.

If we take into account \overline{W} _{(μ_{1},μ_{2})}^{∗} = \overline{W}_{(−μ_{2},−μ_{1})}, then we obtain

\overline{K}_{λ}^{0} = \overline{W}_{(−λ_{2},−λ_{1})} , \overline{K}_{λ}^{1} = \overline{W}_{(λ_{2}+2,−λ_{1})}
\overline{K}_{λ}^{2} = \overline{W}_{(λ_{1}+3,1−λ_{2})} , \overline{K}_{λ}^{3} = \overline{W}_{(λ_{1}+3,λ_{2}+3)} .

\underline{Example}: The case (λ_{1}, λ_{2}) = (0, 0) corresponds to \overline{K}_{λ}^{p} = Ω_{X}^{p}(log) = \overline{Ω}_{X}^{p}.

The central character of λ can also be described by a number 2r ∈ ZZ. For fixed λ_{1}, λ_{2} the possible values of 2r are the integers, such that 2r = λ_{1} + λ_{2} modulo 2. In the case where 2r is even, one can therefore normalize r by twisting with a suitable character to be r = 0. This implies, that with this normalisation scheme the corresponding l-adic sheaves are pure of weight zero and self dual. We refer to this normalisation as the ”unitary” normalisation. It implies, that the central elements of the Hecke algebra act trivially. So all automorphic representations occuring in the cohomology have trivial central character on an open subgroup of Z_{G}(A A)/Z_{G}(Q), hence are unitary of finite order. If 2r is odd, this ”unitary” normalisation scheme can be introduced only in a purely formal manner, and allows to deal with representations of Sp(4, Q) instead of considering GSp(4, Q). See also [C-F], p.231 concerning the slight inconveniance of having thus half integral F-weights in Hodge filtrations. However, as this can be done purely formally and presents no difficulties, we will use this although it yields half integral Hodge structures resp. Weil sheaves. We will most of the time use this ”unitary” normalisation scheme, as it seems to be conveniant when dealing with the trace formula. If one studies Hodge structure or Galois representations on etale cohomology groups, this artificial normalisation should be removed most naturally by an ”r-fold” formal Tate twist (r ∈ \frac{1}{2}ZZ).

Let us first review certain facts about the mixed Hodge structure on H^{3}(X, E_{λ_{1},λ_{2}}). For further details see [C-F], thm. 5.5 and 6.2. The pure part of this Hodge structure, which contains the image of the space of holomorphic cusp forms or the image of the cohomology with compact support, has formal Hodge types (3 + \frac{1}{2}(λ_{1} + λ_{2}), −\frac{1}{2}(λ_{1} + λ_{2})),

(2+\frac{1}{2}(λ_{1}−λ_{2}), 1−\frac{1}{2}(λ_{1}−λ_{2})), (1−\frac{1}{2}(λ_{1}−λ_{2}), 2+\frac{1}{2}(λ_{1}−λ_{2})), and (−\frac{1}{2}(λ_{1}+λ_{2}), 3+\frac{1}{2}(λ_{1}+λ_{2})). This

means, that H_{!}^{3}(X, E_{λ_{1},λ_{2}}) = image(H_{c}^{3}(X, E_{λ_{1},λ_{2}}) → H^{3}(X, E_{λ_{1},λ_{2}})) carries (after removing our formal Tate twist) a pure Hodge structure of weight k_{1} + k_{2} − 3

V = H_{!}^{3}(X, E_{λ_{1},λ_{2}}(−r)) , r = (λ_{1} + λ_{2})/2
V = V^{(k_{1}+k_{2}−3,0)} ⊕ V^{(k_{1}−1,k_{2}−2)} ⊕ V^{(k_{2}−2,k_{1}−1)} ⊕ V^{(0,k_{1}+k_{2}−3)}

provided E_{λ_{1},λ_{2}} is unitarily normalized.

Let us return to the analytic situation. Remember, that the space of global sections H^{0}(\overline{X}, \overline{W}_{(k_{1},k_{2})}) can be identified with the space [Γ, (k_{1}, k_{2})] of holomorphic vector valued modular forms of weight (k_{1}, k_{2}). The degeneration of the spectral sequence [C-F], thm. 5.5 gives inclusions

[Γ, (λ_{1} + 3, λ_{2} + 3)] ↪ H^{3}(X, E_{(λ_{1},λ_{2})}) .

Hence vector valued holomorphic modular forms of weight k_{1} ≥ k_{2} ≥ 3 can be embedded into the cohomology of suitable canonical coefficient systems E_{λ}. These embeddings are compatible with the action of Hecke operators. The case k_{1} = k_{2} = k corresponds to classical Siegel modular forms of weight k.

The space of holomorphic cusp forms [Γ, (k_{1}, k_{2})] maps to V^{(k_{1}+k_{2}−3,0)} for k_{1} ≥ k_{2} ≥ 3.

\underline{Remark}: More generally suppose k_{1} ≥ k_{2} > 0 one could ask, which holomorphic cusp forms can be detected by higher cohomology groups. We check the list above and see, that the space of vector valued holomorphic modular forms of type (k_{1}, k_{2}) can be embedded into the cohomology of a canonical coefficient system iff ρ + (−k_{2}, −k_{1}) is not Sp(4)-singular. In other words in all cases, where k_{2} ≠ 2 and (k_{1}, k_{2}) ≠ (2, 1) resp.

(k_{1}, k_{2}) ≠ (1, 1).

\underline{Example}: For λ = (2, 0) we obtain H^{1}(X, E_{λ}) = H^{1}(\overline{X}, Ω_{X}(log)^{∗}).

2.Automorphic representations:

Consider the group G = GSp(4, Q) of symplectic similitudes and the Shimura varieties S_{K}(G) as in [W]. As analytic spaces they are isomorphic to disjoint unions of Siegel modular varieties as above and the previous considerations carry over. So we can consider X = S_{K}(G) and H^{•}(S_{K}(G), E_{λ}) together with the embeddings of spaces of holomorphic modular forms. The adele group GSp(4, A A_{fin}) acts on the direct limit H^{•}(S(G), E_{λ}) = lim_{K} H^{•}(S_{K}(G), E_{λ}) and defines an admissable representation. The automorphic representations on the spaces of holomorphic vector valued modular forms of type (k_{1}, k_{2}) with k_{1} ≥ k_{2} ≥ 3 (holomorphic discrete series) embed as GSp(A A_{fin}) constituents of the representation H^{3}(S(G), E_{k_{1}−3,k_{2}−3}), as explained above.

It is well known (see [B] or again [CF], p.233), that the subspace of cusp forms in the space of holomorphic modular forms maps under these embeddings into

H_{!}^{3}(S_{K}(G), E_{λ}) = Image(H_{c}^{3}(S_{K}(G), E_{λ}) → H^{3}(S_{K}(G), E_{λ})) .

Fix the type k_{1} ≥ k_{2} ≥ 3, put λ = (k_{1} − 3, k_{2} − 3) and E = E_{λ}. Consider the holomorphic discrete series representation π_{∞} of GSp(4, R) of type (k_{1}, k_{2}) and some fixed irreducible cuspidal automorphic representation π of GSp(4, A A) where π ≅ π_{fin}⊗π_{∞}. Let S be a finite set of places including the ramified places of π_{fin} and the archimedian place. Choose a compact open small level group K ⊂ G(A A_{fin}), such that π^{K} is nonzero. Choose a K such that K = K^{S}K_{S}, where K^{S} is the maximal compact subgroup of unimodular symplectic similitudes outside S.

The Shimura schemes X = S_{K}(G) have canonical models over Q. There are two choices of interest. The first one is the following: Within the set S of bad nonarchimedian places we assume K_{S} to be a ”modified” principal congruence subgroup K_{n}^{(2)} as defined in [W], p.102. This however may be an additional assumption on π! The second choice - the full principal congruence subgroup - avoids this: If we choose the full congruence subgroup on the other hand, the corresponding Shimura scheme, still defined over Spec(Q), is not geometrically connected any more. The space considered with respect to the modified principal congruence group is a geometric quotient. We will consider certain ample divisor on the quotient space, which pull back to an ample divisor of the nonconnected scheme.

Furthermore, for any place p outside of S, there is a smooth canonical model X over Spec(ZZ_{p}) in the sense of Milne.

The Hecke algebra H^{S} of K^{S} biinvariant functions on G(A A^{S}) with compact support acts on H^{•}(S_{K}(G), E) and H_{c}^{•}(S_{K}(G), E) via correspondences. Let us fix further notations. We may choose some nonarchimedian place p not in S. Then we consider the tensor decomposition

π = π_{∞} ⊗ π_{fin} , π_{fin} = π_{S} ⊗ π_{p} ⊗ π_{f} .

In the set of nonarchimedian places we always let indicate S some fixed finite set including all ramified places and p some distinguished unramified place (not in S). At the remaining nonarchimedian places the spherical representation π_{f} is then uniquely determined by the eigenvalues of the action of the Hecke algebra H_{f} on its spherical vector. In the following we consider the generalized eigenspaces H^{•}(S_{K}(G), E(π_{f})), H_{c}^{•}(S_{K}(G), E(π_{f}))

and H_{!}^{•}(S_{K}(G), E(π_{f})).

3.CAP and toric representations:

For the moment let G be any reductive group over Q and π be any irreducible cuspidal automorphic representation on G(A A). For any parabolic subgroup P ⊂ G and almost all places v the following holds

• There exists an irreducible representation τ_{v} of the Levi component of P_{v}, such that π_{v} is a subquotient of Ind_{P_{v}}^{G_{v}}(τ_{v}).

\underline{Definition}: The representation π will be called CAP (see [PS]) resp. toric, if for one choice of a proper parabolic subgroup P ⊂ G over Q the representation ⊗_{v}τ_{v} of the Levi group can be chosen to be automorphic resp. its central characters ω_{v} is automorphic (viewed as a character of the center of the Levi component).

\underline{Remark}: These notions refer to the weak equivalence class of the representation π. Hence we will note distinguish between π and π_{f} in the following. (Two irreducible automorphic representations are weakly equivalent, if they are isomorphic locally at almost all places).

Conjecturally both notions should coincide, i.e an irreducible cuspidal automorphic representation should be toric iff it is a CAP representation. A partial result in case of the group GSp(4) can be found in [W2]. The CAP representations for GSp(4) have been classified by Piateskii-Shapiro and Soudry. See [PS] and [S].

Theorem: Let us mentain the assumptions and notations from above. Let π be an irreducible cuspidal automorphic representation, whose infinite component π_{∞} belongs to a holomorphic discrete series representation of type (k_{1}, k_{2}). Define π_{f} as above. Then

1) One has nontrivial embeddings of π_{f} into H_{!}^{3}(S_{K}(G), E(π_{f})) ⊂ H^{3}(S_{K}(G), E(π_{f})). If π is not CAP (or toric), then

(H^{3}(S_{K}(G), E)/H_{!}^{3}(S_{K}(G), E))(π_{f}) = 0 .

2) All generalized eigenspaces

H^{i}(S_{K}(G), E(π_{f})) = 0 , i ≠ 3

vanish in degree different from i = 3, provided π is not CAP.

Corollary 1: If π is not toric, we get for all Hecke operators f_{p} ∈ C_{c}^{∞}(G(Q_{p})//K_{p})

T race(π_{p}(f_{p}); H^{3}(S_{K}(G), E(π_{f}))) = − ∑_{i=0}^{6}(−1)^{i}T race(π_{p}(f_{p}); H^{i}(S_{K}(G), E(π_{f})) .

\underline{Proof}: We use a slight modification of the arguments in [W1] and [W2]. Especially we use the weak form of the Ramanujan conjecture proved in [CF], p.264. For the following fix some auxiliary prime l, say contained in S. Furthermore fix an isomorphism ι between the algebraic closure of Q_{l} and C. The comparison isomorphism allows to identify Betti cohomology and \overline{Q}_{l}-adic cohomology. We proceed as follows:

By assumption π is not CAP. This implies, that all generalized eigenspaces H^{•}(S, E)(π_{f})) are contained in the subspaces H_{!}^{•}(S, E). This follows from the theory of Eisenstein cohomology and involves considerations on the Borel-Serre compactification (see e.g. [Schw]). We skip the details. By the last remark it is enough to study the cohomology groups H_{!}^{i}(S_{K}(G), E). In other words, one has to show vanishing of H_{!}^{i}(S_{K}(G), E(π_{f})) for all i ≠ 3. This will be done in the remaining sections.

4. The l-adic sheaves:

The next remark concerns the sheaves E. The sheaves E can already be defined as l-adic smooth Weil sheaf on the algebraic variety X = S_{K}(G), considered as a scheme over \overline{Q} and defined over Q. They are of geometric origin and occur in the direct images of products of the universal abelian variety π : A^{s} → S_{K}(G), A^{s} = A ×_{S_{K}(G)} A ×_{S_{K}(G)} ... × A. The basic example is the sheaf R^{1}π_{∗}(\overline{Q}_{l}), which corresponds to the locally constant system E_{λ} for λ = (1, 0). All the other canonical irreducible locally constant coefficient systems E_{λ} can be produced out of such direct images in a similar way. See [CF], p.235.

In order to simplify considerations, we choose a square root p^{1/2} of p in \overline{Q}_{l} and normalize the Weil sheaves E_{λ} to be ι-pure Weil sheaves of weight 0. For example E_{(1,0)} = R^{1}π_{∗}(\overline{Q})_{l}(\frac{1}{2}). This sheaf is selfdual as a Weil sheaf. The same remains true for the other E_{λ} by tensor calculus. If λ_{1} + λ_{2} is even, then this brute device is more elegantly replaced by using a representation λ of GSp(4, Q) with trivial central character, which has the parameter λ_{1} and λ_{2}. Observe, that the central character is given by an r-power, where

r ∈ ZZ and r ≡ λ_{1} + λ_{2} mod 2.

The same argument also works over Spec(ZZ) \ S. Consider a prime not in S and let T be the spectrum of the strict henselization of ZZ with respect to this prime. There exist a smooth model π : X_{T} → T together with the universal abelian variety A (or products of it over X_{T}), which are smooth over X_{T}.

Lemma 1: The construction above defines smooth sheaves E on X_{T}, such that the sheaves R^{•}π_{!}(E) and R^{•}π_{∗}(E) are smooth on T and commute with arbitrary base change

T′ → T.

\underline{Proof of lemma 1}: Observe, that the Leray spectral sequence degenerates at the E_{2}-level by the Liebermann trick. This reduces us to consider the case of the constant sheaf \overline{Q}_{l} and its higher direct images with respect to the morphism f : A^{s} → T. This map is smooth and admits a proper smooth extension \overline{f} : \overline{A^{s}} → T, where the boundary divisor is a relative divisor with normal crossing ([CF], p.195). Then according to [SGA 4\frac{1}{2}], thm.finitude (App.1.3.3(ii)) the pair (f, \overline{Q}_{l}) is cohomologically proper. Hence the higher direct image sheaves Rf_{∗}(\overline{Q}_{l}) commute with arbitrary basechange T′ → T. According to [SGA 4\frac{1}{2}], thm.finitude (App.2.4) then the specialisation resp. cospecialisation maps are isomorphisms, because f is smooth and therefore locally acyclic. This implies, that the sheaves R^{i}f_{∗}(\overline{Q}_{l}) are smooth sheaves.

Without restriction of generality the morphism f can be compactified by a projective morphism \overline{f} ([CF], p.207). Therefore the relative Poinare duality theorem applies for the morphism f and yields

Rf_{∗}RHom(K, f^{!}(\overline{Q}_{l})) ≅ RHom(Rf_{!}(K), \overline{Q}_{l}) .

Because f is smooth, we have f^{!}(\overline{Q}_{l}) = \overline{Q}_{l}[2d](d). Here d is the (well defined) relative dimension. This implies, that the complex RHom(Rf_{!}(\overline{Q}_{l}), \overline{Q}_{l}) on T has smooth cohomology sheaves. But Th.finitude.thm 4.3 implies that this complex is quasiisomorphic to D(Rf_{!}(\overline{Q}_{l})). Hence Rf_{!}(\overline{Q}_{l}) ≅ D(D(R_{!}f(\overline{Q}_{l}))). The spectral sequence converging to R^{i}f_{!}(\overline{Q}_{l}) has E_{2}-terms, which are all smooth and the higher Ext sheaves therefore vanish. Hence the R^{i}f_{!}(\overline{Q}_{l}) are smooth sheaves on T.

\underline{Positive characteristics}: Let us fix some prime q different from p and not in S. Our base change assertion allows to consider the situation in characteristic q using base change. Let \overline{X} denote a smooth proper toroidal compactification over the algebraic closure \overline{F}_{q} of the prime field of chracteristic q. Of course, the sheaf E need not extend to a locally constant coefficient system on \overline{X}.

First of all H_{c}^{•}(X, E) is independent whether we consider X = S_{K}(G) to be the Shimura scheme over C or Q or use its smooth reduction mod q, considered as a scheme over \overline{F}_{q}. Namely these spaces will be identified using the specialisation map. This follows from the base change assertions made above. By Poincare duality, the same holds for cohomology with compact support and therefore also for H_{!}^{•}(S_{K}(G), E). These identifications are equivariant with respect to the action of the Hecke correspondences on these cohomology groups.

Over the field \overline{F}_{q} the Frobenius automorphism also acts on these cohomology groups commuting with the action of the Hecke correspondences. The eigenvalues of Frobenius on the spaces H_{!}^{i}(X, E) are pure of weight i. We have to take into account, that our spaces are not compact. Let \overline{E} be the intermediate extension of the sheaf E to the compactification \overline{X} considered in characteristic q in the sense of Bernstein-Beilinson-Deligne. This is a pure selfdual perverse Weil sheaf of weight 0. Its cohomology groups are pure, hence also the subquotients H_{!}^{•}(X, E).

5.Lefschetz maps:

Next consider Lefschetz maps. The hard Lefschetz theorem gives for all r ≥ 0 isomorphisms

L^{r} : H^{3−r}(\overline{S}_{K}(G), \overline{E}) → H^{3+r}(\overline{S}_{K}(G), \overline{E})(r) ,

where L^{r} is the map induced from the iterated cup product with the fundamental class of some ample line bundle L on S = \overline{S}_{K}(G).

We have the corresponding maps on the cohomology groups and cohomology groups with compact support. These we consider both in characteristic q and characteristic zero. In positive characteristic L commutes with the Frobenius action up to some factor (coming from Tate twist).

We have a commutative diagram

H_{c}^{•}(X, E) → H^{•}(\overline{X}, \overline{E})
↓ id  ↓ res
H_{!}^{•}(X, E) ↪ H^{•}(X, E)

compatible with the Lefschetz maps.

Next we show, that L^{r} commutes with the action of Hecke operators. Using the base change and smoothness assumption, the specialisation map is an isomorphism outside S. It is therefore enough to consider the question in characteristic 0. For that choose L as in [W]. Then, as explained in [W] p.104 and 108, the Lefschetz map L^{r} : H_{!}^{3−r}(X, E) → H_{!}^{3+r}(X, E)(r) is defined by the iterated cup product with respect to the translation invariant harmonic twoform on G(R). Hence L commutes with the action of Hecke operators, because it already does on the level of differential forms. We finally obtain the induced maps

L^{r} : H_{!}^{3−r}(X, E(π_{f})) → H_{!}^{3+r}(X, E(π_{f}))(r) .

Lemma 2: Assume, that π_{f} is not CAP. Then the restriction of L^{r} to the generalized π_{f} eigenspace defines an isomorphism

L^{r} : H_{!}^{3−r}(X, E(π_{f})) → H_{!}^{3+r}(X, E(π_{f}))(r) .

\underline{Proof}: Poincare duality applied to the self dual sheaf E on X proves dim H_{!}^{3−r}(X, E) = dim H_{!}^{3+r}(X, E)(r). Use Im(ϕ)∗ ≅ Im(ϕ∗) for linear maps ϕ : V → W between finite dimensional \overline{Q}_{l} vectorspaces. Apply this to the natural map ϕ from cohomology with compact support to ordinary cohomology. Observe that ϕ is selfdual with respect to Poincare duality. Hence it is enough to show injectivity or surjectivity of L^{r}.

At this point of the proof we have to use a transcendental argument. We throw in our knowledge on spectral theory. Namely according to [Schw] every cohomology class has an automorphic representative in our case. For the general case there is the theorem of J.Franke. By our assumption, all noncuspidal representatives are ruled out. This means, that our assumption on π_{f} implies, that the generalized π_{f} eigenspaces are contained in the cuspidal cohomology and therefore its elements have harmonic representatives (with respect to the translation invariant metric), which are rapidly decreasing on the Siegel space H/Γ. However, the natural map from harmonic rapidly decreasing forms to cohomology H^{•}(H/Γ, E) is injective (see [B], p.47). Coefficients are assumed to be complex. The Lefschetz map is realized by iterated cup products with a translation invariant harmonic two form. Hence it preserves harmonic rapidly decreasing form. This shows, under our hypothesis on π_{f}, that the natural map H^{3−r}(X, E) → H^{3+r}(X, E)(r) is injective. By the dimension count it is an isomorphism.

\underline{Remark}: I do not know, whether the map

L^{r} : H_{!}^{3−r}(X, E) → H_{!}^{3+r}(X, E)(r)

defines an isomorphism. This is true for the constant sheaf E = \overline{Q}_{l}. See [W1].

6.Weak Ramanujan:

This being said, we come back to the representation π_{f}. The weak form of the Ramanujan conjecture proved in [CF], p.264 implies, that the graded generalized eigenspace H_{!}^{•}(X, E)(π_{f}), when considered in positive characteristic q, is preserved by the Frobenius action. Furthermore the Frobenius F_{q} can have at most four different eigenvalues on this subspace. Although not stated explicitly there, the possible eigenvalues are given by

α_{0}, α_{0}α_{1}, α_{0}α_{2}, α_{0}α_{1}α_{2} .

These numbers correspond to the ”roots” of the local spinor L-function

L_{q}(π_{f}, s − \frac{3}{2})^{−1} = (1 − α_{0}q^{−s})(1 − α_{0}α_{1}q^{−s})(1 − α_{0}α_{2}q^{−s})(1 − α_{0}α_{1}α_{2}q^{−s})

attached to π_{f} or better π_{q}. Note the shift by \frac{3}{2}. The complex numbers α_{0}, α_{1}, α_{2} are the so called Satake parameters of the local representation attached to π_{f} at the corresponding prime q, up to the fact that α_{0} has to be normalized. The usual unitary normalization corresponds to the shift, which multiplies α_{0} by the factor q^{3/2}. In this normalization scheme (α_{0}q^{−3/2})^{2}α_{1}α_{2} has absolute value one.

7.Low degrees:

Our next remark concerns the cohomology in degree 0, 1, 5, 6. It is well known, that cuspidal representations do not contribute to these degrees, only one dimensional automorphic representations do. See [B], p.40 and p.42. Namely by Poincare duality it is enough to consider the degrees 0 and 1. Only the degree 1 case needs to be considered seriously. The vanishing theorems for continuous L^{2} cohomology show, that the cohomology groups H_{cusp}^{i}(X, E) (complex coefficients) vanish for i < r, where r = 2 is the rank of our symmetric domain. If therefore π_{f} is not CAP, the generalized eigenspaces H^{i}(X, E(π_{f})) etc. vanish in degree i = 0, 1, 5, 6. Hence we can restrict ourselves to discuss the cohomology groups H_{!}^{i}(X, E) in degree 2, 3 and 4.

8.Multiplicity one:

Assume that π_{f} is automorphic, but not CAP and contributes to degree i = 3. Suppose β_{1}, β_{2} where two different eigenvalues on H_{!}^{2}(X, E)(π_{f}) respectively qβ_{1}, qβ_{2} two different eigenvalues on H_{!}^{4}(X, E)(π_{f}), corresponding vice versa under the Lefschetz isomorphism L. By purity they have to be distinct from each other. By the weak Ramanujan conjecture above they have to be all the eigenvalues. This gives a contradiction, because our representation occurs nontrivially in degree 3, where the eigenvalues of Frobenius have a different absolute value of weight 3. Therefore at most one eigenvalue of the Frobenius F_{q} is realized in H_{!}^{2}(X, E). The same applies to H_{!}^{4}(X, E) by the Lefschetz isomorphism.

9.Globalisation:

For a non CAP representation π_{f} we have shown the following: The Frobenius at the prime q has at most one eigenvalue on the π_{f}-isotypic component of the cohomology in degree two.. More generally if the Galois group Gal(\overline{Q} : Q) acts semisimply on H_{!}^{2}(X, E(π_{f})), then the corresponding Galois representation has to be a multiple of a fixed abelian character. The same is true in degree 4.

Let us show semisimplicity. For that one could proceed as follows.

According to [W], p.104 there exists a nice ample divisor \overline{Y} in \overline{X} described below, such that \overline{Y} = Y ∪(\overline{X} \X). This divisor is defined over Q. Furthermore the inclusion composed with the restriction map

H_{!}^{2}(X, E) ↪ H^{2}(X, E) ↪ H^{2}(Y ∩ X, E|Y ∩ X)

is injective. Namely the kernel of the second map is a quotient of H_{c}^{2}(\overline{S} \ \overline{Y} , E), which is zero by the affine vanishing theorem due to Artin and Grothendieck. The divisor \overline{Y} is singular. It is a union of products of two modular curves together with the boundary divisor \overline{X} \X, which is a union of Shioda surfaces. Passing to the normalisation of Y ∩X, one can control the cohomology groups H^{2}(Y ∩ X, E|Y ∩ X) in terms of cohomology groups of modular curves and canonical coefficient systems on them. As in [W] this implies semisimplicity and that all one dimensional Galois representations, which occur in the (l-adic) cohomology, are given by grossen characters, viewed as characters of the absolute Galois group of Q.

Multiplicity one and the argument above imply, that the Galois group acts on H_{!}^{2}(S, E)(π_{f}) via a multiple of some grossen character χ(−1), χ being a Dirichlet character. Similarily in degree 4 we get the corresponding multiple χ(−2).

Therefore assume, that π_{f} is not a CAP representation and contributes to degree three, however not exclusively. Then qχ(q) and q^{2}χ(q) occur as eigenvalues of the Frobenius F_{q} on the generalized eigenspaces H^{•}(X, E(π_{f})). So these two numbers occur among the four ”roots”. The Weyl group acts transitively on the four ”roots” of the local spinor L-function L_{q}(π_{q}, s). Therefore, without restriction of generality α_{0} = qχ(q). Furthermore q^{2}χ(q) can be assumed to be one of the two numbers α_{0}α_{1} and α_{0}α_{1}α_{2} again using the Weyl group. Hence for almost all primes q one of the two following cases holds

Case 1): α_{1} = q  or  Case 2) : α_{1}α_{2} = q .

10.The first case will not occur:

We will show α_{1}α_{2} = q foe almost all primes q, especially the central character ω_{π_{f}} (q) = α_{0} · α_{0}α_{1}α_{2}q^{−3} = qχ(q)q^{2}χ(q)q^{^{−}3}. This implies, that ω_{π} = (χ)^{2} is a square.

To proof this claim, assume to the contrary, that α_{1} = q holds. The unitary central character ω_{π} of π is related to the value ω′(q) = α_{0}^{2}α_{1}α_{2}. By our conventions this character ω′ is a Dirichlet character twisted by \overline{Q}_{l}(−3) and ω′(q) has absolute value q^{3}. Hence α_{0} = qχ(q) and α_{1} = q imply |α_{2}| = 1. On the other hand, at least one eigenvalue of Frobenius occurs in the third cohomology. In the first case 1) therefore either α_{0}α_{2} or α_{0}α_{1}α_{2} is a number of absolute value q^{3/2}, which gives a contradiction. Hence for all

primes q ≠ p, q ∉ S

α_{1}α_{2} = q  and  α_{0} = qχ(q)

holds. The fact, that π_{f} contributes to cohomology in degree together with the Weil conjecture implies, that |α_{0}α_{1}| = q^{3/2}, hence

|α_{1}| = q^{1/2}

and then also

|α_{2}| = q^{1/2} .

11.A contradiction:

Consider the L-series

L^{S}(π_{f}, s − \frac{3}{2}) = ∏_{q∉S} L_{q}(π_{f}, s − \frac{3}{2}) .

The estimates for Satake parameters above imply

Res_{s=3}L^{S}(π_{f} ⊗ χ̃, s − \frac{3}{2}) = c · Res_{s=1}L^{S}(χχ̃, s) , c ≠ 0 ,

for some nonvanishing const c. The partial L-series has a pole for χ̃χ = 1 at s = 1. This implies, that the partial L-series of π ⊗ χ^{−1} has a pole. Furthermore π has central character χ^{2}, and π ⊗ χ^{−1} has trivial central character. By the characterisation of CAP representation of P GSp(4, A A) of [PS], this implies that π ⊗ χ^{−1} is a CAP representation of P GSp(4, A A). Hence π is a CAP representation of GSp(4, A A), a contradiction to our assumption.

Corollary 2: Let π be an irreducible cuspidal automorphic representation with infinite component π_{∞} belonging to the L-packet of the holomorphic discrete series of type (k_{1}, k_{2}). Assume π is not a CAP representation and E = E_{(k_{1}−3,k_{2}−3)}. Then π contributes only to cohomology in degree 3 in the sense that H^{i}(X, E(π_{f})) = 0 for i ≠ 3.

\underline{Remark}: The CAP representations, which appear for the classical holomorphic Siegel cusp of weight k ≥ 3 (k_{1} = k_{2} = k) are given exactly by the Saito-Kurokawa lift. See [PS]. The situation is reversed for weights k = 1, where only the forms studied by Soudry appear. See [W3].

References:

[B] Borel A., Stable cohomology of arithmetic groups II, collected papers

[CF] Faltings G.-Chai C.L., Degeneration of abelian varieties, Ergebnisse der Math. und ihrer Grenzgebiete 3.Folge Band 22 Springer Verlag (1990)

[PS] Piateskii-Shapiro I, On the Saito-Kurokawa lifting, Inv. math. 71 (1983), 309 - 338

[Schw] Schwermer J., On arithmetic quotients of the Siegel upper half space of degree two, Comp. Math. 58 (1986), 233 - 258

[SGA4\frac{1}{2}] Seminaire de geometrie algebrique du Bois-Marie, cohomologie etale, Springer Lecture Notes in Math. 569, (1977)

[S] Soudry D., The CAP representations of GSP(4,A), J. reine. angew. Math. 383 (1988), 87 - 108

[W] Weissauer R. , Differentialformen zu Untergruppen der Siegelschen Modulgruppe zweiten Grades, J. reine angew. Math. 391 (1988), 100 - 156

[W1] Weissauer R., The Picard group of Siegel modular threefolds, J. reine angew. Math. 430 (1992), 179 - 211 [W2] Weissauer R., On certain degenerate eigenvalues of Hecke operators, Comp. Math. 83, (1992), 127 - 145

[W3] Weissauer R., Modular forms of genus 2 and weight 1, Math. Z. 210, (1992), 91 - 96

The Ramanujan Conjecture for Genus two

Siegel modular Forms

(An Application of the Trace Formula)

R.Weissauer

December 1993, revised in November 1994

University of Mannheim

In this note we apply results of local harmonic analysis obtained for the group GSp(4) in [W1-4]. As indicated by Langlands and in more precise form by Kottwitz, such results of local harmonic analysis are one of the ingredients, which are essential for the understanding of the L-functions of Shimura varieties. The main intention is the proof of the Ramanujan conjecture for automorphic cuspidal representations π on GSp(4, A A), whose infinite component π_{∞} is of discrete series type. This includes the case of vector valued holomorphic Siegel cusp forms of genus two. It is well known that the Ramanujan conjecture does not hold in general. It fails to hold on the subspace defined by the Saito-Kurokawa lift ([PS]). There is the more general class of CAP representations, where the failure of the Ramanujan conjecture has been established. The CAP representations for the group GSp(4) are well understood by the work of Piateski-Shapiro and Soudry [S]. One should believe, that at least for representations whose archimedian components are discrete series representations, being CAP should be the only reason for the failure of the generalized Ramanujan conjecture. Our main result is, that this is true for the group GSp(4): The Ramanujan conjecture (or purity conjecture) holds for all irreducible cuspidal non CAP represenations, with archimedian component in the discrete series.

1.Preliminaries

In this paper we use trace computations due to Kottwitz [K]. We therefore follow the conventions of this paper. Kottwitz works with Shimura varieties S_{K} attached to the structure data (G, h), whose complex valued points are given by S_{K}(C) = G(Q) \ (X_{∞} × (G(A A_{f})/K)), where X_{∞} = G_{∞}/Zentr(h). In [CAP] we used the following variant of this definition. Let K_{∞} denote a maximal compact subgroup in G_{∞} and put K̃_{∞} = Z_{G,∞}^{0}K_{∞}. Then X_{G} = G_{∞}/K̃_{∞} is a connected symmetric space attached to the adjoint group and a corresponding space S_{K} can be defined using X_{G} instead of X_{∞}. There is a finite covering map X_{∞} → X_{G}. The cardinality of the fibers is equal to the index [K̃_{∞} : Zentr(h)]. In the standard case G = GSp(4) and the standard choice of structure morphism h (see also [W]), we get [K̃_{∞} : Zentr(h)] = 2, because K̃_{∞} is generated by Zentr(h) and the block diagonal matrix ϵ = diag(id, −id). To compare results one has to keep in mind the current meaning of the definition of S_{K}. This means, that the trace formulas in [CAP] have to be multiplied by this index, which is two in the case above.

Independently of the definition used, one has canonical coefficient system E_{λ} on these spaces. They are essentially indexed by a pair (λ_{1}, λ_{2}) of integers λ_{1} ≥ λ_{2} ≥ 0 using the notation from [Lef]. On has furthermore to specify the central character, which corresponds to some integer r ∈ ZZ, such that r ≡ λ_{1} + λ_{2} mod 2. In the following we will mainly be interested in the case λ_{1} = λ_{2}, in view of the results of [Lef], [Deg] mentioned above. We will then always assume r = 0 without restriction of generality. It should be said, that for most of the following considerations, this restriction λ_{1} = λ_{2} however is not essential, it is simply a matter of conveniance.

Classical holomorphic Siegel cusp forms of weight k ≥ 3 are related to these coefficient system by generalized Eichler integrals. They define embeddings of the space of holomorphic modular forms as above into third cohomology groups H^{3}(S_{K}, E_{λ}), where k and λ are related by λ_{1} = λ_{2} = k − 3. More generally vectorvalued holomorphic cusp forms of weights k_{1} ≥ k_{2} ≥ 3 correspond to the choice λ_{i} = k_{i} − 3. See [CF] and also [Lef].

Some further constants of relevance are the number of discrete series representations d(G) in a fixed L-packet for the group G_{∞}. For G = Sp(2g, Q) this number is 2^{g}, for G = GSp(2g, Q) it is 2^{g−1}. Especially d(Gl(2, Q)) = 1, hence we also have d(H) = 1 for the group H = Gl(2, Q)^{2}/G_{m}. The group G_{m} is embedded via t ↦ (t · id, t^{−1} · id).

For the standard homomorphism h (chosen e.g. in [Pic]) we have [K̃_{∞} : Zentr(h)] = 2 for G = Gl(2, Q) and more generally G = GSp(2g, Q), as explained above. Therefore the corresponding index is at most 4 in the case of the group M, where h is the induced structure homomorphism \overline{h} : S → Gl(2, R)^{2} → M(R). However the two homomorphisms

a + ib ∈ C^{∗} ↦ ( \begin{matrix}a & b \\ −b & a \end{matrix} )

a + ib ∈ C^{∗} ↦ ( \begin{matrix}a & −b \\ b & a \end{matrix} ) = Ad(ϵ)(( \begin{matrix}a & b \\ −b & a \end{matrix} ))

from S → Gl(2, R) mod Z_{Gl(2)}(R) are different even modulo the center of Gl(2, R). Therefore [K̃_{M} : Zentr(\overline{h})] = 4. But one should keep in mind, that the derived group attached to M is \underline{not} simply connected!

We are mainly interested in the standard case G = GSp(2g, Q), where the Tamagawa measure is

τ(G) = 1 .

This is clear from the general formula τ(G) = |π_{0}(Z(Ǧ))^{Γ}|/|ker^{1}(Q, Z(Ǧ))|, valid for all reductive groups whose derived group are simply connected. For the groups G = GSp(2g, Q) we have Z(Ǧ) = C^{∗}, considered with trivial Galois action.

In the case g = 2, we also have to consider the group M = Gl(2, Q) × Gl(2, Q)/Q^{∗}. Here the subgroup Q^{∗} is embedded via the map x ↦ (x · id, x^{−1} · id). The derived group M_{der} is Sl(2, Q) × Sl(2, Q)/(±1) is not simply connected. However use of [Oe], p.40 f shows that

τ(M) = τ(M)τ(Q^{∗})/τ(Gl(2, Q)^{2}) is equal to the product of #Kokern(Hom(Gl(2, Q)^{2}, G_{m}) → Hom(Q^{∗}, G_{m})) and [M(A A)/M(Q)Gl(2, A A)^{2}] divided by

#Kern(kern^{1}(Q, G_{m}) → Kern^{1}(Q, Gl(2)^{2})) .

The first factor is two and the second and third vanish by Hilbert theorem 90. Thus

τ(M) = 2 .

Finally there is the constant

i(G, M) = τ(G)τ(M)^{−1}λ^{−1} ,

where λ is the order of a group Λ(M, s, η) = Aut(M, s, η)/M_{ad}(Q) attached to the elliptic endoscopic group M. Aut(M, s, η) is the group of those automorphisms of M, which preserve the embedding η up to Ǧ equivalence and the element s under the induced map K(M/Q) → K(M/Q). See [Ko], p.630 for further details. Up to equivalence there exist a unique nontrivial elliptic endoscopic triple (M, s, η) for G = GSp(4, Q)

η : M̌ = \{( \begin{matrix}∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \\ ∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \end{matrix} )\} ↪ Ǧ = GSp(4, C)

for the choice

s = diag(1, −1, 1, −1) .

Any automorphism ρ of P Gl(2) is inner. Therefore an automorphism ρ of M, which does not switch the two factors of M_{ad} can be modified by an inner one, such that it induces the identity on M_{ad}. ρ preserves the center, which is isomorphic to G_{m}. Hence the restriction ρ_{0} of ρ to the center Z_{M} = G_{m} of M is either the identity or the inversion on Z_{M} = G_{m}. The homomorphisms of ρ : M → M, which induce the identity on M_{ad} and some given homomorphism ρ_{0} on the center Z_{M} correspond to characters ψ ∈ Hom(M, Z_{M}), which induce ψ_{0}(z) = ρ_{0}(z)z^{−1} on Z_{M}. If ρ_{0} = id then follows ρ = id, because Hom(M/Z_{M}, Z_{M}) is trivial. In our case the group Hom(M, Z_{M}) is isomorphic to the character group of M, which is generated by (g_{1}, g_{2})/(t · id, t^{−1} · id) ↦ det(g_{1}g_{2})^{−1}. The center Z_{M} is generated by the equivalence classes of elements (id, t · id)/ ∼. The

map (g_{1}, g_{2})/ ∼↦ (g_{1}, g_{2}det(g_{1}g_{2})^{−1})/ ∼= (g_{1}det(g_{1})^{−1}, g_{2}det(g_{2})^{−1}) is an automorphism of

M, which is not inner and induces the inversion on Z_{M}. But its dual is not induced by an inner automorphism of Ǧ. There it does not contribute to Λ(M, s, η). Modulo inner automorphism there is a unique further automorphism of M given by (g_{1}, g_{2})/ ∼↦ (g_{2}^{−t}, g_{1}^{−t})/ ∼, which switches the factors of M_{ad}. The induced map on M̌ is the restriction of an inner automorphism of G, which stabilizes s modulo the center of Z(Ǧ). Therefore λ = 2. We obtain

i(G, M) = \frac{1}{4} .

2.Results of Kottwitz, Milne and Pink, Shpiz

Now consider a prime p of good reduction for the Siegel modular variety S_{K} under consideration. Consider compact open subgroups K = K_{p}K^{p} in the groups over the finite adeles, where K_{p} = GSp(ZZ_{p}). In [K] Kottwitz states a conjectural formula (3.1) for the supertrace of a correspondence

f × F rob_{p}^{n}

on the etale cohomology groups H_{c}^{•}(S_{K}(\overline{F_{p}}), E_{λ}). The sheaves E_{λ} are certain canonical smooth l-adic sheaves on the reduction of the Shimura variety, temporarily viewed as a variety over the algebraic closure \overline{F_{p}} of the prime field of characteristic p. See [Lef]. The Hecke operator f = ∏_{v} f_{v} in the Kottwitz formula is a product of local Hecke operators coming from the primes outside p. There is no restriction up to the fact, that f = f_{p}f^{p} is K-biinvariant and induces at the place p the local Heckeoperator f_{p}, which is the unit element of the local Hecke algebra. For the more precise definitions see loc.cit.

In principal the supertrace T race_{s} = T race_{s}(f × F rob_{p}^{n}) can be calculated by the Lefschetz trace formula, due to Grothendieck-Verdier. However its local terms are complicated and there are possible boundary contributions due to the fact, that the variety is not proper over the base field. On the other hand R.Pink has shown in [P], that at least for a high enough power n >> 0 of the Frobenius, the supertrace of the correspondence can be calculated by a more naive form of the Lefschetz trace formula, where the local terms are suggested by a simple counting argument. However exactly by the heuristically use of this ”naive” form of the Lefschetz trace formula, Kottwitz derived his conjectural formula. In the special case of the standard Shimura varieties attached to the symplectic group of similitudes G = GSp(2g, Q) with the standard homomorphism, Kottwitz was able to verify the ”naive” kind of the Lefschetz trace formula [K], (3.1). This is explained in part III of [K] in the case of trivial Hecke operator and trivial coefficient system. In Milne [M1,M2] one finds an extension to the case of arbitrary coefficient systems and arbitrary Hecke operators outside p.

Hence, by the results of Pink, Kottwitz and Milne the formula is truly valid for G = GSp(2g, Q) for high enough Frobenius twists (n >> 0).

3.Stabilisation

Furthermore Kottwitz was able to rearrange the summation terms in the formula (3.1), using certain standard conjectures on (twisted) endoscopy, such that it appears in the stabilized form

T race_{s}(f × F rob_{p}^{n}) = ∑_{E} i(G, M)ST_{e}^{∗}(h) .

This needs some explanations.

1) Firstly, E refers to a set of representatives of elliptic endoscopic triples. For example, in the case of the group G = GSp(4, Q), the cardinality of E is two. There is the trivial triple (G, 1, id) and the triple (M, s, η) considered above.

2) Secondly, the constants i(G, M) are certain rational numbers. In our case they are 1 and \frac{1}{4} respectively.

3) Thirdly, the symbol ST_{e}^{∗}( ) stands for the (G, M)-regular Q-elliptic part of the stable trace formula.

ST_{e}^{∗}(h) = τ(M) ∑_{γ_{M}} |(M_{γ_{M}} /M_{γ_{M}}^{0} )(Q)|^{−1}SO_{γ_{M}}^{M} (h) .

The sum runs over a set of representatives for the (G, M)-regular semisimple Q-elliptic stable conjugacy classes in M(Q). The terms SO_{γ_{M}}^{M} (h) are the corresponding stable orbital integrals.

Finally the functions h = ∏_{v} h_{v} are carefully attached to the given Hecke operator. This is done by local constructions, which for every fixed elliptic endoscopic datum, denoted M for simplicity, attaches to each f_{v} a corresponding function h_{v} = f_{v}^{M}. This is described in [K] modulo some local conjectures on endoscopic transfer.

Let us specialize to the case G = GSp(4, Q). Modulo the local conjectures on the endoscopic lifts, which in this case are proved in the papers [W1], [W2], [W3] and [W4] we get the formula

T race_{s}(f × F rob_{p}^{n}; H_{c}^{•}(\overline{S_{K}}, E_{λ})) = ST_{e}^{∗}(h^{G}) + i(G, M)ST_{e}^{∗}(h^{M}) .

Let us consider its first term on the right hand side. It is

ST_{e}^{∗}(h^{G}) = τ(G) · ∑_{γ_{G}} SO_{γ_{G}}^{G}(h^{G}) .

Summation is over all semisimple Q-elliptic elements in the stable conjugacy classes of G(Q). The condition of G − G regularity is void.

4.Destabilisation

Results of Kottwitz [Ko] on the stabilisation of the elliptic part of the trace formula allows to destabilize this formula (again modulo certain endoscopic conjectures, which are covered by the results in the case G = GSp(4, Q) mentioned above). This gives

ST_{e}^{∗∗}(h^{G}) = T_{e}^{∗}(h^{G}) − i(G, M)ST_{e}^{∗∗}((h^{G})^{M}) .

Here, following the notation of Kottwitz, the double index ∗∗ instead of ∗ should remind us, that in the case G = M one should not consider the G − M regular elements, but exclude the central elements. We need not go through the full prozess of destabilisation to observe, that the second term vanishes. This follows, because it is a product over local stable orbital integrals. These are, up to some transfer factors, equal to the so called κ-orbital integrals of the functions h^{G}. For the vanishing of the product of local κ-orbital integrals it is enough to know, that at the archimedian place we have

h_{∞}^{G} = (−1)^{q(G)}d(G)^{−1} ∑_{π∈Π_{disc}(τ)} f_{π_{∞}} .

This is a sum over certain pseudo coefficients f_{π_{∞}} of discrete series representations π_{∞} attached to an irreducible (finite dimensional) representation τ of the compact form corresponding to the coefficient system E_{λ} under consideration. This is explained in [K], section 7. Such an average h_{∞}^{G} of pseudo coefficients has trivial κ-orbital integral at the archimedian place. Hence the whole second sum cancels. We get

ST_{e}^{∗∗}(h^{G}) = T_{e}^{∗}(h^{G}) ,

where the remaining term is the elliptic trace

T_{e}^{∗}(h^{G}) = ∑_{γ_{G}} |(G_{γ_{G}}/G_{γ_{G}}^{0})(Q)|^{−1}τ(G_{γ_{G}}^{0})O_{γ_{G}}^{G}(h^{G}) .

This sum runs over all ordinary conjugacy classses of semisimple Q-elliptic elements in G(Q), however with central elements removed. See [Ko] p.391 ff, especially theorem 9.6. Again it should be said, that all orbital integrals above are defined via the canonical measures on G(A A) resp. G_{γ_{G}}^{0}(A A) (i.e volumes on the adele quotients are equal to the Tamagawa numbers).

Now add the central terms. In the stable trace ST_{e}^{∗}(h^{G}) they are τ(G)h^{G}(γ_{G}), and also in the unstable trace case they are given by τ(G)h^{G}(γ_{G}). This implies

ST_{e}^{∗}(h^{G}) = ∑_{γ_{G}} |(G_{γ_{G}}/G_{γ_{G}}^{0})(Q)|^{−1}τ(G_{γ_{G}}^{0})O_{γ_{G}}^{G}(h^{G}) .

Now summation is over all conjugacy classes of semisimple Q-elliptic elements in G(Q). However, the orbital integrals over h_{∞}^{G} at the archimedian place are zero unless the elements are R-elliptic. This allows us to compare this formula with the terms of the topological trace formula of Goresky and MacPherson.

5.The topological trace formula

For an adelic version of the topological trace formula of Goresky-Macpherson and some simplifications related to representations, which are not CAP, we refer to [CAP]. In the following, all measures are again supposed to be canonical measures. Assume, that the level group K is small in the sense of [CAP]. The function h_{∞}^{G} mentioned above coincide with the function f_{∞} in the notation of [CAP]. Let us choose a Hecke operator f, such that it defines a good π-projector for an irreducible automorphic representation π of GSp(4, A A) in the sense of [CAP], constructed with respect to the fixed nonarchimedian places different from p and the primes (in some finite set S) of bad reduction determined by the level subgroup. If we choose K to be a principal congruence subgroup, the operators we consider are in the Hecke algebra H_{S,K}, defined in the appendix of [CAP]. It is easy to see that the assumptions A1, A2 and A3 of [CAP] are satisfied. If we assume, that π is not CAP in the sense of Piateskii-Shapiro, we get from [CAP]

ST_{e}^{∗}(h^{G}) = (d(G)[K̃_{G,∞} : Zentr(h)])^{−1} ∑_{ν}(−1)^{ν}T race(h_{p}^{G}, H^{ν}(S_{K}(G), E_{λ})(π_{f}))

= \frac{1}{4} · ∑_{ν}(−1)^{ν}T race(h_{p}^{G}, H^{ν}(S_{K}(G), E_{λ})(π_{f})) .

This needs some explanations. First observe, the functions h_{v}^{G} were constructed in [K] such that h_{v}^{G} = f_{v} for all v not in S ∪ {p, ∞}. At the archimedian place its definition has already been given. At the primes in S the functions h_{v}^{G} are the characteristic functions of the level group K_{v}, suitable normalized. Normalisation can be omitted if the canonical measure on G(A A) is chosen such that vol_{G(A A_{fin})}(K) = 1. This we can assume as in [K] and [CAP]. The remaining function h_{p}^{G} depends on the twist n and can be understood in terms of cyclic basechange of degree n. See [K] or [K2]. We will go into the details later. For the moment it is enough to remember the dependence of h_{p}^{G} on j. So we will also write h_{p}^{G} = h_{p}^{(n)} in the following. On the other hand we have yet to explain the meaning of the symbol ...(π_{f}) in the topological trace formula stated above. It means the following. If we decompose the representation π into local components, we get

π = ( ∏_{v∈S∪{p,∞}} π_{v}) ⊗ π_{f} .

Hence π_{f} is an irreducible representation of G(A A_{f}), where A A_{f} = ∏^{,} Q_{v}, where v runs over almost all primes omitting precisely p and the primes in S. We can assume π^{K} ≠ 0 without restriction of generality. This means that π_{p} and π_{f} are spherical representations. They are determined by the corresponding eigenvalues of local spherical Hecke operators. The index π_{f} then indicates the generalized eigenspace of the Hecke algebra outside p and S with respect to this eigenvalue system.

On the other hand, the choice of a good π-projector (π not CAP) also localizes the etale trace formula, considered earlier, with respect to the corresponding generalized eigenspace. We obtain the following formula

\underline{Lemma 1}: Let G = GSp(4, Q) and let π be non CAP. Let K be a fixed small level subgroup with corresponding set S of primes of bad reduction. Then for n >> 0 the Kottwitz functions h_{p}^{G} = h_{p}^{(n)} define spherical Hecke operators, such that for the geometric Frobenius F rob_{p}

4 · T race_{s}(F rob_{p}^{n}; H_{c,et}^{•}(S_{K}(\overline{F_{p}}), E_{λ})(π_{f})) − T race_{s}(h_{p}^{(n)}; H_{Betti}^{•}(S_{K}(C), E_{λ})(π_{f}))
= ST_{e}^{∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) .

\underline{Proof}: Multiply the identity above by 4 and observe, that the two coefficients were both equal to \frac{1}{4}. For that, see our introductory remarks on constants.

\underline{Remark}: As explained in [Lef], our assumption that π is not CAP implies, that all generalized π_{f}-eigenspaces of the cohomology groups (with or without compact support) with respect to π_{f} can be replaced by the corresponding spaces H_{!}^{•}(S_{K}, E_{λ})(π_{f}), where H_{!}^{ν}(S_{K}, E_{λ}) is the image of H_{c}^{ν}(S_{K}, E_{λ}) in H^{ν}(S_{K}, E_{λ}). This shows, that we might drop the index c and replace both cohomology groups by the cohomology groups H_{!}^{ν}(S_{K}, E_{λ})(π_{f}).

\underline{Corollary 1}: Under the assumptions of lemma 1 above we obtain

4 · T race_{s}(F rob_{p}^{n}; H_{!,et}^{•}(S_{K}(\overline{F_{p}}, E_{λ})(π_{f})) − T race_{s}(h_{p}^{(n)}; H_{!}^{•}(S_{K}(C), E_{λ})(π_{f}))
= ST_{e}^{∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) .

As explained in [Lef], the cohomology groups of the corollary in characteristic zero and p can be identified by the choice of an isomorphism ι : \overline{Q}_{l} ≅ C.

6. G-M regularity removed

For further applications of corollary 1, it remains to simplify the right hand side of the formula. For that, we should first understand the meaning of terms, which were removed by the condition (*), namely the condition of G − M regularity. Observe, that the group M has no nontrivial elliptic endoscopic subgroup. Therefore the notions of conjugacy and stable conjugacy conincide. In order to interpret it in terms of an ordinary stable elliptic trace formula on M it remains to understand the following question: What does it mean for a R-elliptic, semisimple element of M(Q) to be G − M-regular? Of cause it is enough to deal with this over the field R of real numbers.

Consider an elliptic, R-elliptic element in M(R). It is of the form γ = (γ_{1}, γ_{2})/ ∼ for two elements γ_{i} ∈ Gl(2, R). Up to conjugacy the γ_{i} have the form

γ_{i} = ( \begin{matrix}cos(α_{i}) & sin(α_{i}) \\ −sin(α_{i}) & cos(α_{i}) \end{matrix} ) , first case

or

γ_{i} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}cos(α_{i}) & sin(α_{i}) \\ −sin(α_{i}) & cos(α_{i}) \end{matrix} ) , second case .

This, because they can be conjugated into the maximal compact subgroup. In the latter case the characteristic polynomials would be χ(t) = t^{2} − 1. It roots are t = ±1 and by further conjugation we can assume α_{i} = 0 in the second cases. Now assume, that both matrices are of the first kind. Put t_{i} = cos(α_{i}) + \sqrt{−1} · sin(α_{i}). By [W1] the condition, not to be G − M regular, means

x = x^{σ} or x^{,} = (x^{,})^{σ} ,

where x = t_{1}t_{2} and where x^{,} = t_{1}(t_{2})^{σ}. Here σ denotes the nontrivial R-automorphism of C. In other words: t_{1}/t_{1}^{σ} is either t_{2}/t_{2}^{σ} or its inverse. In terms of the angle this is 2α_{1} = ±2α_{2} modulo 2π. I claim, that in all the cases the relevant term

SO_{γ}^{M}(h_{∞}^{M}) = 0 , (γ not G-M regular)

vanishes.

Let us first recall the definition of the function h_{∞}^{M} on M(R). This is explained in [K], section 7. Remember that M is adjoint type. This implies d(M) = 1. Each packet of discrete series representations of M(R) is a singleton. Furthermore q(G) = 3. Therefore, according to [K]

h_{∞}^{M} = ⟨μ_{h}, s⟩ · ∑_{ω_{∗}∈Ω_{M} \Ω} det(ω_{∗}(ϕ_{M}))(−1)^{q(G)}h(ϕ_{M})

or in our case

h_{∞}^{M} = −⟨μ_{h}, s⟩ · ∑_{ω_{∗}∈{id,w}} det(ω_{∗}(ϕ_{M}), X^{∗}(T))h(ϕ_{M}) .

The term in front of the sum is a sign factor. Furthermore id and w are representatives of the coset of complex Weilgroups Ω_{M} and Ω of M and G respectively. w interchanges the two factors of the group M. If ϕ_{M} : W_{R} → ^{L}M is a Langlands parameter for M, it gives rise to a (singleton) packet of discrete series representations π_{M} on M(R). Then h(π_{M}) denotes its pseudocoefficient (no average in this case). Up to a sign determined later (!) we therefore obtain

±h_{∞}^{M} = h(π_{r_{1}} ⊗ π_{r_{2}}) − h(π_{r_{2}} ⊗ π_{r_{1}}) , (r_{1} > r_{2}) .

Here, by abuse of notation, we regard the discrete series representations π_{M} under consideration as given by a pair of discrete series representations π_{r_{1}}, π_{r_{2}} on the group Gl(2, R). Their weights r_{1} and r_{2} (highest K-types) are numbers ≥ 2 determined by the assumption, that the composite map W_{R} → ^{L}M → ^{L}G gives the discrete series packet on G(R) corresponding to τ, i.e to the coefficient system E_{λ} under consideration. For example, if E_{λ} corresponds to classical Siegel modular forms of weight k, which means λ_{1} = λ_{2} = k−3 (see [Lef]), then we get r_{1} = 2k−2 and r_{2} = 2. See also [L]. More generally for λ_{1} ≥ λ_{2} ≥ 0 and k_{i} = λ_{3} + 3 with k_{1} ≥ k_{2} ≥ 3, we get r_{1} = k_{1} + k_{2} − 2 and r_{2} = k_{1} − k_{2} + 2. Now stable orbital integrals on M(R) are essentially products of ordinary orbital integrals on the groups Gl(2, R). In order to show vanishing of SO_{γ}^{M}(h_{∞}^{M}) in the second cases above, it is enough to show

O_{γ_{i}}(h_{π_{r}}) = 0 , γ_{i} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

The functions h_{π_{r}} are pseudocoefficients for discrete series representations for the groups Gl(2, R), which are not connected. To show vanishing of the orbital integral we argue as follows. Let G = G^{+} ∪ G^{−} be the two connected components of Gl(2, R)/Z(R)^{0}. Pseudo coefficients f have to satisfy the following property

∫ f(x)T race π(x)dx = c(π)

with c(π) = 0 for all tempered representations, except those in the discrete series packet of the given representation. Then it should be one. On G^{+} the given discrete series representation π_{r} splits into a package with two elements π_{r}^{+}, π_{r}^{−}, one holomorphic and one antiholomorphic. Let f^{+} be a pseudocoefficient for this package. Extend f^{+} by zero to a function f on G. As the restriction of π_{r} from G to G^{+} decomposes into the two representations π_{r}^{+} and π_{r}^{−}, it is clear that f defines (up to some normalisation factor) a pseudocoefficient for π_{r} on G. We apply this remark. Every element in the conjugacy class of γ_{i} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) does not hit G^{+}. Hence the orbital integral is zero.

So we can assume, that both γ_{i} are of the first case. Then they are contained in an elliptic torus and according to the formula [CAP] the orbital integral is

(−1)^{q(M_{γ}^{0})}d(M_{γ}^{0})/d(M)vol(Z_{M_{γ,∞}}^{0} \ \overline{M}_{γ,∞}^{0})^{−1}(tr(τ_{r_{1}−2} ⊗ τ_{r_{2}−2}) − tr(τ_{r_{2}−2} ⊗ τ_{r_{1}−2}))(γ) .

Here τ_{r} denotes the finite dimensional representation of the ”compact form” \overline{M} of highest weight r. This means that η : \overline{M} → M is an isomorphism over C, such that η^{σ} ∘ η^{−1} is an inner automorphism of M. The R-split components of M and \overline{M} may be identified. \overline{M}(R) divided by the split component is compact. In our case this inner cocycle is given by g ↦ g^{−t}det(g) = wgw^{−1}, where w = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Hence \overline{Gl}(2, R) is the group GU(2) of all complex matrices satisfying g^{t}\overline{g} = det(g) · id. This implies det(g) ∈ R. The special unitary group SU(2) together with the group of real diagonal matrices generates the group\overline{M}. Namely ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}\overline{a} & \overline{c} \\ \overline{b} & \overline{d} \end{matrix} ) is always a matrix, whose first entry is real and positive.

In our special case the irreducible representations τ_{r_{i}−2} have the same central character. We can therefore assume γ ∈ SU(2) without restriction of generality to show

SO_{γ}^{M}(h_{∞}^{M}) = const · (trτ_{r_{1}−2}(α_{1})trτ_{r_{2}−2}(α_{2}) − trτ_{r_{1}−2}(α_{2})trτ_{r_{2}−2}(α_{1})) = 0 .

The last equality follows from the following observation: Put ζ = exp(iα). Then the function trτ_{r}(α) = trτ_{r}(−α) is of the form ζ^{a}P_{r}(ζ^{2}), where P (X) is a polynomial in X and X^{−1} and where a is 0 or 1 and congruent to r. Therefore we can first assume 2α_{1} = 2α_{2}

modulo 2π, hence ζ_{1}^{2} = exp(2iα_{1}) = ζ_{2}^{2} = exp(2iα_{2}). Thus (ζ_{1}ζ_{2})^{a}P_{1}(ζ_{1}^{2}, ζ_{2}^{2}) is symmetric in

ζ_{1} and ζ_{2}, because r_{1} and r_{2} have the same parity a. Therefore SO_{γ}^{M}(h_{∞}^{M})(α_{1}, α_{2}) = 0.

\underline{Corollary 2}: The right hand side in the formula of corollary 1 is equal to the ordinary trace

∑_{γ_{M}}[M_{γ} : M_{γ}^{0}]^{−1}τ(M_{γ}^{0})O_{γ}^{M}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) .

The summation runs over all conjugacy classes of R-elliptic, semisimple elements of

M(Q).

\underline{Proof}: By the arguments above we have

ST_{e}^{∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) = ST_{e}^{∗∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) .

The index (**) indicates, that only central elements of M(Q) are omitted in the summation. By destabilisation we obtain again

ST_{e}^{∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) = T_{e}^{∗}(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) ,

because M does not have nontrivial elliptic endoscopic groups. To the latter sum one can also add the central terms, which are zero because of the presence of the orbital integrals of h_{∞}^{M}. This proves the claim.

7.CAP localisation revisited

At this point we apply the trace formula for M to rewrite everything in terms of automorphic representations of M(A A). We know, that the functions h_{v}^{M} are locally constant with compact support and independent of the twist n. For for the finite set of ramified primes see [Ha]. For the other places, which are unramified, the fundamental lemmas [W1-4] give us precise information on the functions h_{v}^{M}. Unfortunately both the Frobenius trace formula of Kottwitz nor the topological trace formula of Goresky and MacPherson (in the crude adelic form of [CAP]) formally require the derived group to be simply connected. Because of the fact that M_{der} is not simply connected, we go back to the use the the ”ordinary” trace formula. We use it in its L^{2} version, as given in Arthurs paper [A]. This is allowed, because the functions h_{∞}^{M} at the archimedian place are linear combinations of pseudocoefficients of discrete series representations on M. Note, that Arthurs L^{2}-formula can be brought into a CAP localized form in the same way as the topological trace formula was brought, namely the condition A1 of [CAP] is satisfied in our case.

Two things should be said. In the paper [CAP] we have seen, that the trace formula considerably simplifies, if one discards CAP representations in the discrete spectrum.

This can be achieved by inserting suitable testfunctions into the trace formula. These terms where called good projectors in [CAP] In the construction of a good π-projector in [CAP] we specified one place v (not in S ∪ {p}), where the support of the Satake transforms is ”very” regular. In our case of the group M we consider the local Hecke operators h_{p}^{M} and f_{π}^{M}. These functions do not define a good projector on M in the precise meaning of [CAP]. However it can be used in the same way as in [CAP] to localize the trace formula in order to eliminate the parabolic terms. Namely all the function h_{p}^{M} have Satake transforms with regular support for all n >> 0. This will become clear from the precise expression for h_{p}^{M} coming from the twisted endoscopic fundamental lemma, which will be given later. So the prime p takes over the role of the place v in the notation of [CAP]. Secondly, the functions f_{π}^{M} definitely do not project onto a single irreducible representation of M. For instance they can not distinguish representations, which are conjugate under the outer automorphism of M defined by a switch of factors. However they still allow to eliminate the parabolic terms of the trace formula. Namely, again from the explicit description given by the endoscopic fundamental lemma it is clear that any irreducible automorphic representation of M, which is a constituent of a globally induced representation from an automorphic representation of a proper parabolic subgroup of M, is orthogonal to f_{π}^{M}. Irreducible representations of M(A A_{f}) are given by pairs of irreducible automorphic representations π_{1}, π_{2} of Gl(2, A A_{f}), whose central characters ω_{1} = ω_{2} coincide. As f_{π}^{M} is spherical, it is enough for that to consider the case where π_{1} and π_{2} are unramified. If one of the two representations is of Eisenstein type or especially trivial, then the the endoscopic lift of the representation π_{1} ⊗ π_{2} gives an automorphic irreducible spherical representation Π of GSp(4, A A_{f}). From the endoscopic fundamental lemma it is clear, that Π is then induced from an automorphic representation of the Levi subgroup of the Siegel parabolic subgroup. As π was assumed to be not CAP, the π-projector can be chosen such, that it is orthogonal to all such lifts Π. There are only finitely many possibilities for Π, because without restriction of generality the representations π_{1} ⊗ π_{2} can be considered to have fixed discrete series components at the archimedian place and fixed level. Furthermore the level is determined by the functions h_{v}^{M} for v ∈ S, which are not changed for varying n. We get from the L^{2}-trace formula of Arthur for M after the process of CAP localisation, which in the present case obviously also throws away all terms in the L^{2}-cohomology in degree different from two, the

\underline{Corollary 3}: The right hand side of the formula in corollary 1 is equal to

∑_{π_{1}⊗π_{2}} tr(π_{1} ⊗ π_{2})(h_{∞}^{M}h_{p}^{M}f_{π}^{M}) .

The sum runs over all cuspidal representations π_{1} ⊗π_{2} of M(A A) given by pairs of cuspidal automorphic representations of Gl(2, A A), with equal central characters ω_{1} = ω_{2}, such that ω_{1,∞}, restricted to the connected component of the center, is trivial.

\underline{Remark}: The argumentation above especially eliminated all the Saito-Kurokawa lifts.

\underline{Proof of Corollary 3}: Using CAP-localisation for Arthur’s formula [A], Thm. 6.1 for L_{μ} = L_{μ}(h_{p}^{M}f_{π}^{M}) leaves us with the elliptic trace terms of M and eliminates all the complicated parabolic contributions. The remaining trace term can be rewritten in the form

d(M) ∑_{γ∈M(Q)∼} τ(M_{γ}^{0})[M_{γ} : M_{γ}^{0}]^{−1}O_{γ}^{M(A A )}(h_{p}^{M}f_{π}^{M}f_{∞})

where the sum is over all semisimple R-elliptic conjugacy classes in M(Q). f_{∞} = (−1)^{q(M)}d(M)^{−1} ∑ h(μ), where the sum runs over D(M) discrete series pseudo coefficients h(μ). In our case d(M) = 1 and q(M) = 2 we have f_{∞} = h(μ). So a comparison with the right side of corollary 1 gives the the new formula

±(L_{μ_{1}⊗μ_{2}} − L_{μ_{2}⊗μ_{1}})

for the right side of corollary 1. Now by [A], prop 2.1 we have

L_{μ}(h_{p}^{M}f_{π}^{M}) = ∑_{π^{M}} m(π^{M})χ_{μ}(π_{∞}^{M})trπ_{fin}^{M}(h_{p}^{M}f_{π}^{M}) .

The sum runs over all irreducible representations π^{M} of M(A A) in the discrete spectrum of L^{2}(M(Q)Z_{M}(R)^{0} \ M(A A). See [A], p.263. But the multiplicity one theorem for Gl(2) implies m(π^{M}) = 1. Again the trace of f_{π}^{M} eliminates all π^{M} = π_{1} ⊗ π_{2}, where either π_{1} or π_{2} is one dimensional. This uses the fundamental lemma and the fact that f_{π} was a good π-projector of a non CAP representation π. There remain only cuspidal contributions in the summation. The cohomological Euler characteristic χ_{μ}(π_{∞}^{M}) vanishes unless π_{∞}^{M} = π_{r_{1}} ⊗ π_{r_{2}} is the unique discrete series representation of M(R), determined by μ = μ_{1} ⊗ μ_{2}. In the present case of small group M this is simply a matter of checking the relevant cases and in principle is well known. In the nonvanishing case the Euler characteristic is (−1)^{q(M)} = 1 because of the choice of the group K̃ or KR^{′} in the notations of [A] resp. [CAP]. See [A], lemma 2.2. We therefore get for the right side of corollary 1 the expression

∑_{π^{M}} trπ^{M}(h_{p}^{M}f_{π}^{M}h_{∞}^{M}) .

Summation is over all π^{M} = π_{1} ⊗ π_{2}, where π_{1}, π_{2} run over the isomorphism classes of irreducible cuspidal automorphic representations of Gl(2, A A), such that central characters ω_{1} = ω_{2} coincide. Furthermore, because of the pseudo coefficients which enter in h_{∞}^{M}, and our normalisation of the representation λ, we can furthermore assume that the archimedian component of the ω_{i} has order two depending on the parity of the r_{i}.

Let us look at the formula of the last corollary in more detail. As

±h_{∞}^{M} = h(π_{r_{1}} ⊗ π_{r_{2}}) − h(π_{r_{2}} ⊗ π_{r_{1}}) ,

only those irreducible cuspidal automorphic representations π_{1}, π_{2} of Gl(2, A A) need to be considered, whose archimedian component is discrete series of weight either r_{1} or r_{2} with archimedian central character sign^{r_{i}}, by our normalisation. We can therefore assume, that π_{1} and π_{2} have unitary central character.

8. Applying the fundamental lemma

Remember the definition of the functions f_{π}^{M}. For given f_{π} on GSp(4, A A_{f}), the function f_{π}^{M} can be any locally constant function with compact support on M(A A_{f}) satisfying

Δ(η, (t_{1}, t_{2}))O_{η}^{κ}(f_{π}) = SO_{(t_{1},t_{2})}^{M}(f_{π}^{M}) .

We refer to [W1-2] for the notations above. Especially (t_{1}, t_{2}) is a pair of Gl(2)- representatives of a semisimple G − M regular element in M(A A_{f}). η is a corresponding element in GSp(4, A A_{f}). For the precise definition of η we refer to [W1-2].

Let us see what happens, if we interchange t_{1} and t_{2}. In case 2 (loc.cit.) the element η and the transfer factor depend only on the product t_{1}t_{2} and is therefore a priori symmetric. In case 1 (loc.cit.) this is not a priori clear. The interchange of t_{1} and t_{2} corresponds to the change of variables x, x′ ↦ x, (x′)^{σ}. This changes η in its stable conjugacy class by the ”factor” χ_{L/F} (−1). Furthermore Δ, which depends only on the coordinates x, x′ changes by Δ(x, (x′)^{σ}) = χ_{L/F} (−1)Δ(x, x′). This means, that the right hand side of the identity above is in both cases invariant under the interchange of the variables t_{1} and t_{2}. This interchange induces an outer automorphism of M, which is trivial on the center of M.

In other words, if f_{π}^{M} satisfies the identity above, then also the new function obtained from the twist by the outer automorphism (g_{1}, g_{2}) ↦ (g_{2}, g_{1}) of M. If we replace f_{π}^{M} by its arithmetic mean, we can assume that the function f_{π}^{M} is symmetric under this automorphism.

The functions h_{p}^{H} and h_{∞}^{H}, however, are not symmetric. They are antisymmetric. For h_{∞}^{M} this is evident from the formula above. For h_{p}^{M} the antisymmetry follows from Kottwitz’s description. If we simultaniosly change s to −s and η by precomposing with the outer automorphism, which switches the factors, then h_{p}^{M} does not change. So the effect of the outer automorphism is the same as the change of s by −s. However, changing s by −1 ∈ Z(Ĝ) changes h_{p}^{M} by the factor μ_{1}(−1)^{−1} = −1 by the remarks of [K] following forrmula (7.3).

For the unramified place this alternatively follows from the twisted endoscopic fundamental lemma, proved in [W4]. Namly by the description of [K] (see also [W4]) the twisted endoscopic fundamental lemma gives

tr(π_{1,p} ⊗ π_{2,p})(h_{p}^{M}) = tr π_{1,p}(f^{(n)}) − tr π_{2,p}(f^{(n)}) ,

where f^{(n)} is the analog for Gl(2, Q_{p}) of the spherical Hecke operator h_{p}^{(n)} for GSp(4, Q_{p}), up to a normalisation factor. Without restriction of generality π_{1,p} and π_{2,p} therefore are spherical. More precisely in terms of Satake parameters we have the following picture. Let π_{1,p} and π_{2,p} have parameters α, β resp. α′, β′ with αβ = α′β′, then the trace of h_{p}^{M} on π_{1,p} ⊗ π_{2,p} is

p^{3n/2}((α)^{n} + (β)^{n} − (α′)^{n} − (β′)^{n}) .

The endoscopic lift π_{p} is an unramified representation described in terms of Satake parameters by α_{0} = α, α_{0}α_{1}α_{2} = β, α_{0}α_{1} = α′ and α_{0}α_{2} = β′.

Observe, that the symplectic Satake parameters are determined by those of M and vice versa. However, what really counts are the orbits under the Weyl groups. Therefore two unramified representations π_{1,p} ⊗ π_{2,p} and π′_{1,p} ⊗ π′_{2,p} of M_{p} have the same symplectic unramified lift π_{p} iff they are conjugate under the outer automorphism, which switches the two factors of M.

Finally we give the trace of h_{p}^{G} = h_{p}^{(n)} on the spherical representation π_{p} with the Satake parameters α_{0}, α_{1}, α_{2}. It is

p^{3n/2}((α_{0})^{n} + (α_{0}α_{1}α_{2})^{n} + (α_{0}α_{1})^{n} + (α_{0}α_{2})^{n}) .

The last statement follows from [K2], Thm. 2.1.3 and formula 2.2.1.

\underline{Remark}: In the formulas above we used Satake parameters of spherical representations with ”unitary” normalisation, whereas in [Lef] we used another normalisation. To get the normalisation of [Lef] one has to replace the parameter α_{0} above by α_{0}p^{3/2}. This ties in with the normalisation factor p^{3n/2} appearing in the formulas above. We here prefer the unitary normalisation, because the generalized Ramanujan conjecture takes the form |α_{0}| = |α_{1}| = |α_{2}| = 1 in this normalisation.

Let us put things together. As both the contributions from the place p and the place ∞ are antisymmetric and all the other contributions are symmetric we can assume, by switching the two factors of M, that

π_{1,∞} = π_{r_{1}}

π_{2,∞} = π_{r_{2}}

Remember, the weights r_{1} = k_{1} + k_{2} − 2 and r_{2} = k_{1} − k_{2} + 2 satisfy r_{1} − r_{2} = 2(k_{2} − 2) ≥ 2, hence are different. So from this reordering we obtain every summand twice, which gives a factor two in front. Furthermore the trace of pseudo coefficients gives tr(π_{r_{1}}⊗π_{r_{2}})(h_{∞}^{M}) = ±1. The sign will be given later.

\underline{Corollary 4}: The right hand side in the formula of corollary 1 is equal to

±2 · ∑_{π^{M} =π_{1}⊗π_{2}} tr ⊗_{v≠p,∞} π_{v}^{M}(f_{π}^{M}))·(tr(π_{1,p}(f_{p}^{(n)})) − tr(π_{2,p}(f_{p}^{(n)}))) .

The summation is over all cuspidal irreducible automorphic representations π_{1} and π_{2} of Gl(2, A A) with equal central characters

1) ω_{1} = ω_{2}
2) π_{1,∞} = π_{r_{1}} and π_{2,∞} = π_{r_{2}} with fixed r_{1}, r_{2} normalized as above.

\underline{Remark}: One may ask, how many representations π^{M} contribute to the sum for fixed choice of a good π-projector f_{π}. If there is a contribution at all, we call π_{f} a (weak) M-lift or a (weak) lift from π_{1} ⊗ π_{2} , otherwise we say that π_{f} is not an M-lift. To analyse this, check the orbits of Weyl groups! Suppose π^{M} = π_{1} ⊗π_{2} and π^{′M} = π′_{1} ⊗π′_{2} are two cuspidal automorphic representations, that contribute. Then locally for all unramified places either π_{1,v} ≅ π′_{1,v} and π_{2,v} ≅ π′_{2,v} holds or in reversed order π_{1,v} ≅ π′_{2,v} , π_{2,v} ≅ π′_{1,v}. This implies, that both pairs of global representations π_{1} and π_{2} coincide up to a permutation. Suppose this were not true. Then there exists a global pair π′_{1} ⊗ π′_{2}, such that π′_{1} is not isomorphic to π_{1} or π_{2}. Now consider the partial L-series (outside the finite set S)

L^{S}((π′_{1} ⊕ π′_{2}) ⊗ (π′_{1})^{∗}, s)/L^{S}((π_{1} ⊕ π_{2}) ⊗ (π′_{1})^{∗}, s) .

This quotient of partial L-series with the finitely many factors from some finite set S omitted, is one by assumption. However at s = 1 the numerator has a pole and the denominator not. This gives a contradiction. This is at least true, if the finite number of bad factors in S and the archimedian place do not spoil the picture. In our case considered above, where π_{i} and π′_{i} correspond to holomorphic elliptic modular forms, this argument can easily be made rigorous using the fact, that the strong Ramanujan conjecture was proved by Deligne.

\underline{Question}: The local tensor L-function

L_{v}(π_{1} ⊗ π_{2}, s)^{−1} = (1 − α_{1}α_{2}p^{−s})(1 − α_{1}β_{2}p^{−s})(1 − β_{1}α_{2}p^{−s})(1 − β_{1}β_{2}p^{−s})

is invariant under permutation of π_{1,v} and π_{2,v}. Does the corresponding global L-function determine π_{1} and π_{2} up to a permutation?

Let us return to the formula in corollary 4. It holds for all n >> 0. Assume π_{f} is automorphic but not CAP. Then we know from [Lef], that the π_{f} isotypic component is concentrated in the middle cohomology of degree 3. By our ”unitary” normalisation E_{λ} is a Weil sheaf, which is pure of weight 0. Therefore the module H_{!,et}^{3}(S_{K}(\overline{F}_{p}, E_{λ}) is pure of weight 3 by the Weil conjectures. Furthermore by Deligne’s proof of the Ramanujan conjecture for holomorphic discrete series representations in case of Gl(2), we obtain from corollary 4 that p^{−3n/2}tr(h_{p}^{(n)}, H_{!}^{•}(S_{K}(C), E_{λ})(π_{f})) is a finite linear combination of powers of complex numbers of absolute value 1. This implies the following: Every representation π_{p} of GSp(4, Q_{p}) in this module, which by the way is automatically spherical, has Satake parameters α_{0}, α_{0}α_{1}, α_{0}α_{2} and α_{0}α_{1}α_{2} of absolute value 1. We obtain

\underline{Theorem}: Let π = ⊗′π_{v} be an irreducible cuspidal automorphic representation of GSp(4, A A). Assume π is not of type CAP. Furthermore assume π_{∞} is a discrete series representation. Then for all places, where π_{v} is spherical, the generalized Ramanujan conjecture holds. In other words, the corresponding Satake parameters α_{0,v}, α_{1,v} and α_{2,v} of the representation π_{v} are complex numbers of absolute value one.

\underline{Remark}: The proof uses the endoscopic and the twisted endoscopic lemma (for every n >> 0). It is a result anounced by Hales, that the endoscopic fundamental lemma follows from the special case of unit elements at almost all places, proved in the present case in [W1-3]. As this result seems not been published so far, we give a proof in [W4] including the twisted endoscopic case in our special situation by applying the method of Labesse and Clozel ([La],[Cl]).

9.Odds and ends

Let us consider π as in the last theorem and let us assume, that π is not a weak M-lift in the sense above. Then the right side of the formula in corollary 1 vanishes. Write π = π_{p} ⊗ π^{p}, where p temporarily denoted some fixed chosen spherical place. Assume that there is another cuspidal automorphic representation π′ of GSp(4, A A), such that π′ = π′_{p} ⊗ π^{p}. So outside p both representations are assumed to be isomorphic. If π′_{p} is also spherical, then it also contributes to the formula from corollary 1

(∗) 4 · T race_{s}(F rob_{p}^{n}; H_{!,et}(S_{K}(\overline{F}_{p}), E_{λ})(π^{p,∞})) = ∑_{π′} m(π′)tr π′_{p}(h_{p}^{(n)}) .

The sum runs over all irreducible automorphic representations π′ counted with some multiplicities m(π′), such that π′_{v} ≅ π_{v} for v ≠ p, ∞ (and π′_{p} is unramified). Here m(π′) is the multiplicity of π′_{p} in H_{!}(S_{K}(C), E_{λ})(π^{p,∞}).

If we now vary p, we see that for almost all p the previously given representation π is uniquely determined by π^{p}. This is clear from the finiteness of the number of automorphic representation for given type at infinity and fixed level. So for almost all primes p the right side of the formula (*) is just the single term m(π)trπ_{p}(h_{p}^{(n)}).

Let us consider the \overline{Q}_{l}-adic Galois representation of Gal(\overline{Q} : Q) on the isotypic space of cohomology considered. Its semisimplification ρ_{l} is determined by the knowledge of Frobenius traces at almost all places p by the generalized Tchebotarev density theorem. So let us vary the prime p for the moment. The finiteness theorems imply, that for almost all p the sum in (*) involves only one representation π′, where we assume π′_{p} to be unramified. So we get for all p large enough

4 · trρ_{l}(F rob_{p}^{n}) = m(π_{p})trπ_{p}(h_{p}^{(n)} .

This holds for all large n, therefore for all integers n. The formula implies, that p^{−3/2}F rob_{p} has the eigenvalues α_{p,0}, α_{p,0}α_{p,1}, α_{0}α_{p,2} and α_{p,0}α_{p,1}α_{p,2} each with multiplicity m(π). Hence

(∗∗) det(1 − t · ρ_{l}(p^{−3/2}F rob_{p})) = L_{p}(π_{p}, t)^{−m(π)}
L_{p}(π_{p}, t)^{−1} = (1 − α_{p,0}t)(1 − α_{p,0}α_{p,1}t)(1 − α_{p,0}α_{p,2}t)(1 − α_{p,0}α_{p,1}α_{p,2}t) .

Especially we see from this, that the semisimplification ρ_{l} of the Galois representation depends only on the weak (or near) equivalence class of π.

So this raises the question, whether this Galois representation is irreducible and of dimension four. The latter means, that the multiplicity m(π) is one. This is expected.

References

[A] Arthur J., The L^{2}-Lefschetz numbers of Hecke operators, Inv. Math. 97 (1989) , 257 - 290

[Cl] Clozel L., The fundamental lemma for stable base change, Duke Math. J., vol 61 (1990), 255 - 302

[CF] Chai C.L. - Faltings G., Degenerations of Abelian Varieties, Ergebn. d. Math. u. Grenzgeb., 3. Folge Band 22 (1990)

[Ha] Hales T.C., Shalika germs on GSp(4), asterisque 171-172, p. 195 - 256 (1989)

[K] Kottwitz R.E., Shimura varieties and λ-adic representations, Ann Arbor Proceedings (L.Clozel and J.S.Milne), July 1988 in perspectives of mathematics, academic press 1990, 161 - 209

[Ko] Kottwitz R.E., Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 629 - 646

[K2] Kottwitz R.E., Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), 287 - 300

[La] Labesse J.P., Fonctions elementaires et lemma fondamental pour le changement de base stable, Duke Math. J. vol. 61, (1990), 519 - 530

[L] Langlands R., Automorphic representations, Shimura varieties, and motives. Ein M”rchen, Proc. of Symp. in Pure Math. vol. 33 (1979), part 2, 205 - 246

[M1] Milne J.S., The points on a Shimura variety modulo a prime of good reduction, in: The zeta functions of Picard modular surfaces, edited by R.P.Langlands and D. Ramakrishnan, les publicationes CRM (1992), 151 - 253

[M2] Milne J.S., The conjecture of Langlands and Rapoport for Siegel modular varieties, Bull. Amer. Math. Soc. 24 (1991), 335 - 341

[Oe] Oesterle J., Nombres de Tamagawa et groupes unipotentes en characteristique p, Inv. Math. 78 (1984), 13 - 88

[P] Pink R., On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, preprint 1990

[PS] Piateski-Shapiro I.I., On the Saito-Kurokawa Lifting, Inv. Math. 71, 309 - 338 (1983)

[S] Soudry D., The CAP representations of GSp(4, A A), Crelle 391 (1988)

[Shp] Shpiz E., Deligne’s conjecture in the constant coefficient case, thesis 1990 [W1-4] Weissauer R., A special case of the fundamental lemma part I,II,III,IV preprints Mannheim 1993-94

[Pic] Weissauer R., The Picard group of Siegel modular threefolds, Crelle 430 (1992)

[CAP] Weissauer R., CAP-localisation, preprint Mannheim 1993 [Lef] Weissauer R., An application of the hard Lefschetz theorem, preprint mannheim 1993

[Deg] Weissauer R., On certain degenerate eigenvalues of Hecke operators, Comp. Math. 83 (1992), 127 - 145

Character identities and Galois representations

related to the group GSp(4)

Rainer Weissauer, Mannheim

May 1994, revised and expanded in August 1995

University of Mannheim

These notes continue the study of the trace formula in the case of the group GSp(4) started in [W]. Especially we will use the same notations as in [W] without further comment.

One of the main purposes of these notes is to demonstrate, how one can use the Cebotarev density theorem to improve the results obtained from the simple version of the Lefschetz trace formula used in [W]. The unpleasant effect that certain primes have to be deleted from the consideration, may sometimes be related to deeper problems, like for instance the fact that the strong multiplicity one theorem fails for automorphic forms on GSp(4). The technical problems related to this are unpleasant. Via the l-adic representations one can get rid of part of the technical restrictions. Namely the fact, that we were at one particular prime p (the Frobenius prime) not allowed to study additional Hecke operators, can be improved a posteriori.

In the second part of the paper we obtain results on the form of the local endoscopic character lift. We give local character formulas and relate the irreducible characters in the resulting packets to those of theta lifts, as studied in among others in [HPS].

Let me add, that part of the problems considered in [W] and in this paper were considered independently by Harder [Har2] and Laumon [Lau].

For the following let us fix a cuspidal irreducible representation π = π_{∞}π_{fin} of GSp(4, A A), which is not CAP. Fix a principal congruence level subgroup K ⊂ G(A A_{fin}), such that π^{K} ≠ 0 and assume, that π_{∞} is a member in the L-packet of discrete series representations attached to some local coefficient system E of the Shimura variety S_{K}(G), corresponding to the choice λ_{1} = k_{1} − 3, λ_{2} = k_{2} − 3 ≥ 0 in the sense of [Lef]. Here k_{1} ≥ k_{2} ≥ 3.

1.Lifts

Assume, that π is not CAP but a weak cuspidal lift attached to some fixed irreducible automorphic cuspidal representation σ of M(A A) = Gl(2, A A)^{2}/A A^{∗}. This assumption will be mentained for the whole paper, with one exception at the end of section 4. The representation σ is uniquely determined by its lift π, if we normalize σ_{∞,1} and σ_{∞,2} to be discrete series corresponding to holomorphic forms of weight r_{1} = k_{1} + k_{2} − 2 and r_{2} = k_{1} − k_{2} + 2 (note r_{1} > r_{2} ≥ 2) respectively. This is a consequence of the strong multiplicity one theorem for Gl(2, A A). Conversely the representation π need not be uniquely determined by σ. Let us remind the technical definition for π_{f} being a weak lift of some σ_{f}. For this implicitly there was some fixed finite set of nonarchimedian places S containing the level primes of the group K. Furthermore one had to specify some distinguished prime p not in S. Let us call p the Frobenius prime. Then we constructed a good π^{S} projector. The use of the trace formula and the endoscopic fundamental lemma then suggested to relate π_{f} and σ_{f} if σ_{v} = σ_{v,1} ⊗ σ_{v,2} is necessarily unramified with Satake parameters α, β of σ_{v,1} and α′, β′ of σ_{v,2}, whenever π_{v} is unramified and

L_{v}(π_{v}, s) = L_{v}(σ_{v,1}, s)L_{v}(σ_{v,2}, s) , (v ∉ S ∪ {p})

holds for all v ≠ p not in S. So if this holds, we say π_{f} is a weak lift of σ_{f}.

\underline{Remark}: S in the definition above is some fixed set and causes no trouble. However, one wants to vary the prime p in order to study the Galois action of Gal(\overline{Q}/Q) on the generalized eigenspaces of ⊗_{v∉S}π_{v}. A priori the definition of weak lift could depend on the prime p chosen. We will show however, that this is not the case.

\underline{Remark 2}: If π_{f} is a weak lift of σ_{f}, then σ = σ(π) is completely determined by π_{f} or even ⊗_{v∉S}π_{v} for some arbitrarily large finite set of places S. But there may be different choices of π with the properties above, which are releated to the same σ due to a possible failure of the strong multiplicity one theorem for GSp(4). However we will show, that the local component π_{v} of a weak lift π_{f} of some σ_{f} is uniquely determined by σ_{v}, if σ_{v} is spherical. More precise information is then given in section 10, using the classification of irreducible admissable representations of GSp(4, F) for local fields F.

To clarify this, let us temporarily still fix K and the archimedian discrete series packet corresponding to k ≥ 3 as above. Hence we will only consider representations irreducible automorphic representations π′, satisfying π^{′K} ≠ 0 and have archimedian component π′_{∞} in the fixed discrete series L-packet. This is a finite set R of equivalence classes of representations π′. If π, π′ ∈ R are two weak lifts of the same representation σ, then of course π_{v} ≅ π′_{v} holds for all v ∉ S and v ≠ p. A seemingly confusing point is the dependency of the notion of weak lift on the choice of the Frobenius prime p, which is fixed so far. We want to be able to vary p. For that observe, that for primes p sufficantly large π_{p} is the unique unramified representation extending ⊗_{v≠∞,p}π_{v} to an irreducible automorphic representation occuring in R. For simplicity and in order to fix ideas we will therefore enlarge S, so that every prime p outside S will have this property. Of course K and k are still fixed. Hence we get π^{′S} ≅ π^{S} for all π′ in R.

Thus, emphazising the role played by S, we will consider all irreducible representations π′ of GSp(4, A A), such that

π′ ≅ π′_{∞} ⊗ π′_{S} ⊗ π^{S} ,

where π′_{∞} is in the discrete series L-packet determined by k and where π′_{v} ≅ π_{v} holds for all nonarchimedian places v ∉ S. This set of equivalence classes of representations - temporarily ignoring their multiplicities - is called the \underline{weak} \underline{S-equivalence} \underline{class} of π^{S} and the weight k.

2. Galois representations:

Assume, that π^{S} is a weak lift from the endoscopic group M. Our main object of study will be the finite dimensional \overline{Q_{l}} vectorspace

V = H_{!}^{3}(S_{K}(G), E)(π^{S}) .

On this space we have an action of the Hecke algebra H(G(A A_{S})//K_{S}) of K_{S} biinvariant functions on G(A A_{S}).

\underline{Question}: For irreducible π_{S} the space π_{S}^{K} is an irreducible module under H(G(A A_{S})//K_{S}). If it is nontrivial, does it then determine π_{S}? This is clear for spherical representations or more generally in the case where K_{S} is a good small subgroup in the sense of [Be], cor. 3.9. They form a cofinal system. If not, let [π_{S}]_{K_{S}} denote the set of equivalence classes of representations π′_{S} determined by this. [π_{S}] consists of those representations π′_{S}, for which π′_{S}^{K_{S}} is isomorphic to π_{S}^{K_{S}} as a Hecke module.

We have a decomposition

V = H_{!}^{3}(S_{K}(G), E)(π^{S}) = ⨁_{π^{S}} m(π_{S})π_{S}^{K} ,

where the sum is over representatives of the weak S-equivalence classes for our fixed π^{S} and k. Only finitely many π_{S} contribute. In the following we replace unitarily normelized coefficient systems E by their natural twists E(−r) as in [Lef].

The action of the absolute Galois group Gal(\overline{Q} : Q) commutes with all Hecke operators. Therefore it respects the decomposition with respect to irreducible representations π_{S} under G(A A_{S}). Consider the modular tower for arbitrary small K’s! As above let m(π_{S}) denote the multiplicity of the constituent π_{S} ⊗ π^{S} in lim_{K}H_{!}^{3}(S_{K}(G), E). Then one has a representation

ρ_{π_{S}} : Gal(\overline{Q} : Q) → Gl(m(π_{S}), \overline{Q}_{l})

for each π_{S}, which arises from some λ-adic representation. We want to determine its semisimplification. This can be of course recognized on the finite level K, as long as

π_{S}^{K} ≠ 0.

We like to relate the representation ρ_{π_{S}} attached to π_{S}π^{S} to the representations ρ_{1}, ρ_{2} attached to σ_{1} and σ_{2}. According to Deligne and Ribet we have two dimensional irreducible \overline{Q}_{l} representations ρ_{1}, ρ_{2}, scalar extensions of two dimensional λ-adic representations

ρ_{λ,1}, ρ_{λ,2}

attached to σ_{1} and σ_{2}. They are characterized by the properties

trρ_{λ,1}(F rob_{p}) = p^{(r_{1}−1)/2}(α_{p} + β_{p}) , detρ_{λ,1}(F rob_{p}) = p^{r_{1}−1}α_{p}β_{p}
trρ_{λ,2}(F rob_{p}) = p^{(r_{2}−1)/2}(α′_{p} + β′_{p}) , detρ_{λ,2}(F rob_{p}) = p^{r_{2}−1}α′_{p}β′_{p}

which holds for all unramified primes p ∉ S. Here we consider geometric Frobenii and the cohomological Galois representations, which are dual to the ones, usually considered.

It turns out, that the Hodge structure of V corresponds to the Hodge type

(V^{(r_{1}−1,0)} ⊕ V^{(0,r_{1}−1)}) ⊕ (V^{(r_{2}−1,0)} ⊕ V^{(0,r_{2}−1)})(−k_{2} + 2) .

Note that (r_{1} −1)−(r_{2} −1) = 2(k_{2} −2). This is a pure Hodge structure of weight k_{1} +k_{2} −3. Due to our (unitary) normalisations we tend to work with the formal Tate twists V (−r) with half integral r = (k_{1} + k_{2} − 3)/2 ∈ \frac{1}{2}ZZ instead of V itself. This was useful from the point of view of automorphic forms. It is not natural from the point of view of Galois representations, and explains the twists introduced in the following. Our plan is to show, that the representation

ρ_{π_{S}} ⊗ ν_{l}^{r}

is related to the representations

ρ_{1} and ρ_{2} ⊗ ν_{l}^{k_{2}−2} .

Here ν_{l} denotes the cyclotomic character ν_{l} : Gal(\overline{Q} : Q) → Q_{l}^{∗} in the sense that ν_{l}(F rob_{p}) = p. This is already strongly indicated by the fact, that the semisimplifications are determined by the characteristic polynomials of the Frobenius elements F rob_{p} for p ∉ S chosen from a set of Dirichlet density one. From the Frobenius relation proved by Chai and Falting [CF] one knows, that the characteristic polynomial det(id − tF rob_{p}; ν_{l}^{r} ⊗ ρ_{π_{S}}) divides a power of

L_{p}(π_{p} ⊗ ν^{r+ \frac{3}{2}} , t)^{−1} = det(id − F rob_{p}t; ρ_{1} ⊗ ν_{l}^{k_{2}−2} ⨁ ρ_{2}) .

This holds for all p ∉ S and follows from the desciption of the endoscopic lift from σ^{S} to π^{S} and the corresponding matching of the local L-functions descibed above. In other words, we have up to the obvious possible permutations under the Weyl groups

α = α_{p,0}
β = α_{p,0}α_{p,1}α_{p,2}
α′ = α_{p,0}α_{p,1}
β′ = α_{p,0}α_{p,2} .

This determines the eigenvalues of the Frobenius elements F rob_{p} for all p ∉ S. They are among those four numbers. It remains to determine their multiplicities. We show the existance of two integers m_{1}(π_{S}), m_{2}(π_{S}) such that

ν_{l}^{r} ⊗ (ρ_{π_{S}})^{ss} = m_{1}(π_{S})ρ_{1} ⨁ m_{2}(π_{S})(ν_{l}^{k_{2}−2} ⊗ ρ_{2})

holds. To achieve this we only need equality of characteristic polynomials of Frobenius for primes p ∉ S in a set of Dirichlet density one.

\underline{Remark}: The set of primes p, such that α_{p} = β_{p} or α′_{p} = β′_{p} is finite. This follows as in [Rib], p.37. Namely let K be the coefficient field. This is a finite extension field of Q. We have to exclude the case a_{p}^{2} = 4p^{k−1}ω(p), where a_{p} = T r(F rob_{p}) ∈ K and where k = r_{1} − 1 or k = r_{2} − 1. This implies, that a_{p} has nonintegral valuation values, i.e. K/Q is ramified at p. This is possible only for finitely many primes p.

\underline{Assumption 1}: Exclude these finitely many primes p for the comparison.

Under this assumption, one finds that the number of different Frobenius eigenvalues is either 2 or 4 depending on whether σ_{p,1} ≅ σ_{p,2} holds or not. If both σ_{p,i} are isomorphic we normalize the Satake parameters, such that α_{p} = α′_{p} and β_{p} = β′_{p}.

\underline{Remark}: Suppose, that not both σ_{i} are of CM-type, then the set of primes p, such that σ_{p,1} ≅ σ_{p,2} has Dirichlet density 0.

This follows from Serre [Se1] theorem 10. This theorem shows that the natural density is zero, hence the analytic Dirichlet density is zero. See also [Se2], p.76. The theorem 10 of loc. cit. is applicable in the following setting. Consider the two dimensional λ-adic representations ρ_{λ,2} and ρ_{λ,1} over some common λ-adic coefficient field. The determinants of F rob_{p} are ω(p)p^{r_{1}−1} and ω(p)p^{r_{2}−1}, because the nebentype character of σ_{1} and σ_{2} coincide. We denoted it ω. Consider the 4-dimensional representation

ρ_{λ,1} ⨁(ρ_{λ,2} ⊗ ν_{l}^{k_{2}−2}) .

A Goursat type argument (see for example [Weiss], p. 121) shows, that either both representations are of CM-type or that ρ_{λ,1} ⊗ κ ≅ ρ_{λ,2} ⊗ ν_{l}^{k_{2}−2} ([Weiss], Behauptung 5.3) for some Dirichlet character κ or that the Galois group has an image, which is a Lie group of dimension greater than the image in each of the two single representations. In the last case theorem 10 of [Se1] applies. The second case cannot occur, because it contradicts the strong multiplicity one theorem. Remember that the weights r_{1} > r_{2} ≥ 2 and are always different! The first case, that σ_{i} are both CM representations with imaginary quadratic CM-fields K_{i}, was excluded.

In the excluded case, where both representations are CM with CM fields K_{1} and K_{2} we have α = −β for all primes inert in K_{1}/Q resp. α′ = −β′ for all primes inert in K_{2}/Q.

Therefore σ_{p,1} ≅ σ_{p,2} holds for all primes, which are simultaniously inert in K_{1}/Q and K_{2}/Q. If K_{1} = K_{2} this is a set of Dirichlet density 1/2. Otherwise consider the third quadratic extensions field K_{3} ⊂ K_{1}K_{2}. Every p, such that F rob_{p} maps to a generator of Gal(K_{1}K_{2}/K_{3}) in Gal(K_{1}K_{2}/Q), induces nontrivial automorphisms in Gal(K_{1}/Q) and Gal(K_{2}/Q). By Cebotarev this set of primes has density 1/4. For the other primes we have only a finite number of conincidences. Namely then we have α_{p} = −β_{p} for all primes inert in the CM extension K/Q. For the split primes p = ℘\overline{℘} we have α_{p} = ψ(℘) and β_{p} = ψ(\overline{℘}) for some grossen character ψ of type (k, 0), where k = r_{1}−1 (resp. k = r_{2}−1 in the case of α′ = ψ′(℘)). So what has to be studied is one of the equalities

ψ(℘) = \{ \begin{matrix}ψ′(℘) \\ ψ′(\overline{℘}) \\ √ & \overline{−ω(p)p^{t}} \end{matrix} .

They all are of the type ρ(℘) = 1, where ρ is some grossencharacter of infinity type (n, m), n ≠ m. This however is impossible. Consider the idele x = (1, ..., ℘, 1, ....) of A A_{K}^{∗}. Then ρ(x) = ρ(℘), where ρ : A A_{K}^{∗}/K^{∗}U → C^{∗} is an idele class character, where U is some compact open subgroup of the finite ideles. Finiteness of class group implies for a suitable power y of x, y = k_{∞}^{−1}ku for u ∈ U and k ∈ K^{∗} with infinite component k_{∞}. Thus ρ(k_{∞}) = (k_{∞})^{n+mσ} = 1. This is only possible if |k_{∞}| = 1, and then means k_{∞}^{n−m} = 1. In other words k_{∞} is a root of unity, because n ≠ m. But then a power of y, which itself is a power of x, lies in U. This is a contradiction, because no positive power of x can be in U!

Now choose an auxiliary unramified prime p such that the following holds

\underline{Assumption 2}: Assume p is chosen in a set of primes such that for the corresponding Satake parameters α ≠ β and α′ ≠ β′ and α ≠ α′, β′ holds. This is a set of primes of positive Dirichlet density. (It is 1 or 1/2 or 1/4 depending on the three cases: not both forms σ_{i} are CM, both are CM with the same CM field or with different CM-fields).

3.The trace formula applied

Let the situation be as in the last section, especially suppose π^{S} to be a weak lift from M. Now we apply the trace formula as in [W], where the auxiliary prime p ∉ S is chosen as in assumption 1 above. We get from [W], corollary 4 the \underline{trace identity}

4 · ∑_{π_{S}} trπ_{S}(f_{S}) · (m_{1}(π_{S})α_{p}^{n} + m_{2}(π_{S})(α′_{p})^{n} + m_{3}(π_{S})β_{p}^{n} + m_{4}(π_{S})(β′_{p})^{n})

− ∑_{π_{S}} tr_{π_{S}}(f_{S}) · (m_{1}(π_{S}) + m_{2}(π_{S}) + m_{3}(π_{S}) + m_{4}(π_{S})) · (α_{p}^{n} + (α′_{p})^{n} + β_{p}^{n} + (β′_{p})^{n})

= ±2 · trσ_{S}(f_{S}^{M})(α^{n} + β^{n} − (α′)^{n} − (β′)^{n}) .

The sum is over all the finitely many π_{S}, which extend to an irreducible automorphic representation π_{∞}π_{S} ⊗π^{S} in the weak S-equivalence class of the σ-lift π considered above (therefore with fixed π^{S} unramified outside S and ∞). Furthermore the function f_{S} is an arbitrary K_{S}-biinvariant function on GSp(4, A A_{S}) with compact support. The integers

m_{i}(π_{S})

are the \underline{multiplicities} of the four possible Frobenius eigenvalues α_{p}, α′_{p}, β_{p}, β′_{p} of p^{−3/2}F rob_{p} on the π_{S} ⊗π^{S}-isotypic space ρ_{π_{S}} of the cohomology V as considered in corollary 1. They may depend on the chosen prime p.

\underline{Remark}: The trace identity above holds for all n >> 0, therefore a posteriori for all n.

From this trace identity and the assumption 2 above together with the partial symmetry of the right side and symmetry of the second term of left side and because of α_{p} ≠ β_{p} resp. α′_{p} ≠ β′_{p} it follows, that

m_{1}(π_{S}^{K}) = ∑_{π_{S}∈[π_{S}]_{K}} m_{1}(π_{S}) = m_{3}(π_{S}^{K}) = ∑_{π_{S}∈[π_{S}]_{K}} m_{3}(π_{S})

resp.

m_{2}(π_{S}^{K}) = ∑_{π_{S}∈[π_{S}]_{K}} m_{2}(π_{S}) = m_{4}(π_{S}^{K}) = ∑_{π_{S}∈[π_{S}]_{K}} m_{4}(π_{S}) .

This follows from the linear independency of a finite number of irreducible characters of the algebra H(G(A A_{S})//K_{S}). This was true under the stronger assumption 2. If we only assume assumption 1, then we get with a similar argument the weaker statement

m_{1}(π_{S}^{K}) + m_{2}(π_{S}^{K}) = m_{3}(π_{S}^{K}) + m_{4}(π_{S}^{K})

in this case. Of course, this is everything we can expect in this case. In any case by assumption 2 or assumption 1 in case σ_{p,1} ≅ σ_{p,2} we get the following: The semisimplification of the Galois action on the submodule

∑_{π_{S}∈[π_{S}]_{K}} m(π_{S})π_{S} = m(π_{S}^{K})π_{S}^{K} ⊂ H^{3}(S_{K}(G), E)(π_{S})

of V is given by

(∗∗) m_{1}(π_{S}^{K})ρ_{1} ⨁ m_{2}(π_{S}^{K})(ρ_{2} ⊗ ν_{l}^{k_{2}−2}) ,

when restricted to the cyclic subgroup generated by each F rob_{p}. If we can show, that the numbers m_{1}(π_{S}^{K}) and m_{2}(π_{S}^{K}) do not depend on the prime p varying, then this essentially gives the global decomposition we are looking for. We show this independence under our assumption 2 on p above. The cases of CM is treated directly. In that case we have already shown, that if the assumption 2 does not hold, but if we exclude a finite number of primes p (assumption 1), that the eigenvalues of F rob_{p} on ρ_{π_{S}} are ±\sqrt{−ω(p)p^{2r+3}}, where each choice of sign of the root comes with equal multiplicity. This equal distribution of signs already follows from the weaker equality m_{1}(π_{S}^{K})+m_{2}(π_{S}^{K}) = m_{3}(π_{S}^{K})+m_{4}(π_{S}^{K}) proved above. This equal distribution of the two signs on the other hand establishes (∗∗) for any

choice of integers m_{1}(π_{S}^{K}), m_{2}(π_{S}^{K}), such that m_{1}(π_{S}^{K}) + m_{2}(π_{S}^{K}) = \frac{1}{2} ∑_{π_{S}∈[π_{S}]_{K}} dim(ρ_{π_{S}}) =

\frac{1}{2}dim(ρ_{π_{S}^{K}}). Hence for the global proof of the decomposition (**) it is enough, to show the independence of m_{1}(π_{S}^{K}) and m_{2}(π_{S}^{K}) under our assumption 2 above.

Inserting the last information m_{1}(π_{S}^{K}) = m_{3}(π_{S}^{K}) and m_{2}(π_{S}^{K}) = m_{4}(π_{S}^{K}), using assumption 2 for the still fixed prime p, we get

∑_{[π_{S}]_{K}} trπ_{S}^{K}(f_{S}) · 2(m_{1}(π_{S}^{K}) − m_{2}(π_{S}^{K}))·(α^{n} + β^{n} − (α′)^{n} − (β′)^{n})

= ±2 · trσ_{S}(f_{S}^{M})·(α^{n} + β^{n} − (α′)^{n} − (β′)^{n}) ,

which holds for all n. Let us put

±n(π_{S}^{K}) = m_{1}(π_{S}^{K}) − m_{2}(π_{S}^{K}) ∈ ZZ .

To solve our problem of independency, it is enough that the numbers n(π_{S}^{K}) are independent from the prime p varying. If this is the case m_{1}(π_{S}^{K}) and m_{2}(π_{S}^{K}) are also independent from p and are the looked for integers in the global decomposition (**). This follows because the number m_{1}(π_{S}^{K}) + m_{2}(π_{S}^{K}) = \frac{1}{2}dimρ_{π_{S}^{K}} = \frac{1}{2}m(π_{S}^{K}) is obviously independent from p.

By assumption 2 the term α^{n} + β^{n} − α^{′n} − β^{′n} does not vanish for all n and this factor can be canceled from both sides of the equality above. Using transfer of characters as in Rogawski [Rog ], p.183 ff we get for the lifted character χ_{σ_{S}}^{G} a finite character expression with integral coefficients and only finitely many irreducible representations π_{S}

χ_{σ_{S}}^{G} = ∑_{π_{S}} n(π_{S}) · χ_{π_{S}} .

This is an identity on the algebra of K_{S} biinvariant functions with compact support on G(A A_{S}). From the uniqueness of character relations for irreducible characters of this algebra, this gives local identities for all v ∈ S

χ_{σ_{v}}^{G} = ∑_{π_{v}} n(π_{v}^{K_{v}}) · χ_{π_{v} ^{K}_{v}} , n(π_{v}^{K_{v}}) ∈ ZZ ,

which recombine into the formula

n(π_{S}^{K}) = ∏_{v∈S} n(π_{v}^{K_{v}}) .

Linear independence of characters of H(G(A A_{S})//K_{S}) therefore shows, that these numbers n(π_{S}^{K}) only depend on the representation π_{S}^{K} and not on the auxiliary primes p chosen. Thus (**) has been proved.

There is one drawback. The character relations obtained are valid only for functions f_{S}, which are K_{S} biinvariant. However we have shown, that they are independent of the auxiliary primes p. Thus at this stage of the discussion we are free to make K_{S} smaller. This happens at the cost of the fact, that now

χ_{σ_{v}}^{G} = ∑_{π_{v}} n(π_{v}) · χ_{π_{v}} , n(π_{v}) ∈ ZZ

is only a countable sum of (unitary) characters! It is an identity of distributions on G(A A_{S}). For each locally constant function with compact support inserted almost all summands are zero. Especially this countable sum is absolute convergent. It is well known, that independence of unitary characters holds in such a setting. See [Rog], prop. 13.8.1. Thus

\underline{Corollary 1}:

n(π_{S}) = ∏_{v∈S} n(π_{v})

m_{1}(π_{S}) − m_{2}(π_{S}) = ±n(π_{S})
ρ_{π_{S}}^{ss} ⊗ ν_{l}^{r} ≅ m_{1}(π_{S})ρ_{1} ⨁ m_{2}(π_{S})ρ_{2} ⊗ ν_{l}^{k_{2}−2} .

Of course, the information encoded in the character identity is already very strong. Let us give an example. The character relations obtained for the set S of bad primes gives a purely local way to understand the ambiguity in the lifting of the representation σ. As we have seen, σ determines π^{S} for each weak lift π. It also determines the possible components π_{v}, v ∈ S. They are in the support of the function n(π_{v}). Similarily it determines the possible π_{S}^{K}. Let us give an immediate application, which shows that our careful preparation of the set S made at the beginning was only a technical construction, which turns out to be unnecessary in the end. The next corollary shows, that one could have chosen S to be the set of all ramified primes of the given representation π.

\underline{Corollary 2}: For given π, K and k let S be the finite set of places constructed as in the beginning. Let π′_{S} = ⨂_{v∈S} π′_{v} be in the weak S-equivalence class of π attached to K and k. For every v ∈ S, which is not a level prime for K and where π′_{v}, v ∈ S is spherical, we have π′_{v} ≅ π_{v}.

\underline{Proof}: Spherical representations π′_{v} of G_{v} are uniquely determined by the module π′_{v}^{K_{v}}, where K_{v} is the corresponding maximal compact subgroup. We have

χ_{σ_{v}}^{G} = ∑ n(π′_{v}^{K_{v}})π′_{v}^{K_{v}} .

This is considered as an identity on the spherical Hecke algebra. On the other hand σ_{v} is spherical. This was a consequence of the endoscopic fundamental lemma. Furthermore

χ_{σ_{v}}^{G}(f) = χ_{σ_{v}}(f^{M}) = tr σ_{v}(f^{M}) = tr π_{v}(f) ,

again by the the endoscopic fundamental lemma. Here π_{v} was assumed to be spherical. Therefore n(π′_{v}^{K_{v}}) = 0 unless π′_{v} ∈ [π_{v}]_{K}. For (spherical) π′_{v} this implies π′_{v} ≅ π_{v} and proves the assertion.

\underline{Remark}: Changing π allows to choose the set S as the set of ramified primes of the representation σ.

\underline{Corollary 3}: The space

lim_{K}H_{!}^{3}(S_{K}(G), E(−r)) = ⨁_{π_{fin}} ρ_{π_{fin}} ⊗ π_{fin}

is an admissable preunitary representation of G(A A_{fin}). Let π = π_{∞}π_{fin} be an irreducible automorphic representation of the group GSp(4, A A). Assume π to be a weak lift: Let S be a finite set of places containing the ramified and archimedian places. Outside S assume L^{S}(π_{S}, s) = L^{S}(σ_{1}, s)L^{S}(σ_{2}, s), where σ_{1}, σ_{2} are two irreducible admissable representations attached to holomorphic elliptic modular forms of weight r_{1} and r_{2} respectively, especially then π is not CAP. Let E be the \overline{Q_{l}}-adic locally constant coefficient system on S_{K}(G), pure of weight zero, attached to the representation with heighest weight (k_{1}−3, k_{2}−3) and assume π_{∞} to belong to the L-packet of the corresponding discrete series representation. Then we have for the semisimplification of the finite dimensional \overline{Q_{l}} representation ρ_{π_{fin}}

ρ_{π_{fin}}^{ss} ⊗ ν_{l}^{r} ≅ m_{2}(π_{fin})(ρ_{2} ⊗ ν_{l}^{k_{2}−2}) ⨁ m_{1}(π_{fin})ρ_{1} .

The integers m_{1}(π_{fin}) and m_{2}(π_{fin}) are related by

m_{1}(π_{fin}) − m_{2}(π_{fin}) = ε ∏_{v≠∞} n(π_{v}) , π_{fin} = ⨂_{v≠∞} π_{v} ,

where n(π_{v}) are integers uniquely determined by the identity

χ_{σ_{v}}^{G} = ∑ n(π_{v})π_{v} .

The sign ε = ±1 is a fixed sign (it is an archimedian contribution corresponding to the normalisation of σ_{∞,1}, σ_{∞,2}). We show later in [W3], that the sign ε is equal to ε = −1. The sum is a countable, absolutely converging sum over the unitary spectrum of GSp(4, Q_{v}). If σ_{v} is spherical, then there exists a unique spherical π_{v}, such that n(π_{v}) ≠ 0. For this we have n(π_{v}) = 1.

\underline{Question}: Is it true that m_{1}(π_{fin}) + m_{2}(π_{fin}) = 1? Probably one has to use the twisted trace formula for Gl(4) × Gl(1) to analyse that question ^{#}.

The occurance of two Hodge types measured by the multiplicities m_{1} and m_{2} (see corollary 5) correspond to the archimedian components of the L-packet

χ_{σ_{∞}}^{G} = π_{+}(σ_{∞}) − π_{−}(σ_{∞}) , n(π_{±}(σ_{∞})) = ±1,

# Note added: This is true and can be proved without using the twisted trace formula.

where π(σ_{∞}) is the non holomorphic discrete series representation and π_{−}(σ_{∞}) is the holomorphic discrete series representation. In other words

m(π_{+}(σ_{∞} ⊗ π_{fin}) = m_{2}(π_{fin})
m(π_{−}(σ_{∞} ⊗ π_{fin}) = m_{1}(π_{fin}) .

If we take this into account a positive answer to the question m_{1}(π_{fin}) + m_{2}(π_{fin}) = 1 is equivalent to the multiplicity formula:

m(π) = \frac{1}{2}(1 + ∏_{v} n(π_{v})) π weak lift of σ , σ generic .

The product is now over all places including the archimedian.

4.Refinements

The Galois representation ρ_{π_{S}} has coefficients of the characteristic Frobenius-polynomials lying in some number field E. This is evident from corollary 1 because the same is true for ρ_{1} and ρ_{2}. This allows to view (**) as an isomorphism of E_{λ}-adic representations. Consider the set Σ_{E} of embeddings τ : E → \overline{Q}. Furthermore consider fixed embeddings of \overline{Q} into C and \overline{Q_{l}} and the completion C_{l}. Note, that the assignments f ↦ f^{M} defined by Hales [Ha] defines the distribution lift α ↦ α^{G}. Then we have from the explicit formula defining χ_{σ_{v}}^{G}, that this is an E-valued function on G_{v}^{e}. Remember, that for a class functions α on M, which transforms with respect to a character ω under the center Z, and a locally constant functions f, compact modulo the center, which transforms according to the character ω^{−1} under Z we have

∫_{G/Z} f(g)α(g)^{G}dg = ∫_{M/Z} f^{M}(h)α(h)dh .

Observe, that the central character may switch during the lifting. Following [Rog], p.182 we will compute α^{G}. First observe, that O_{t}^{M}(f^{M}) = Δ(t, η)O_{η}^{κ}(f) is invariant under the outer automorphism t ↦ t^{∗} of M switching the two factors of M. If t = (t_{1}, t_{2})/(t, t^{−1}) ∈ Gl(2, F)^{2}/F^{∗}, then t^{∗} = (t_{2}, t_{1}). Hence we can assume f^{M}(t^{∗}) = f^{M}(t). The space of class functions α is split into two subspaces, those which are antiinvariant α(t^{∗}) = −α(t) and those which are invariant α(t^{∗}) = α(t). Obviously α^{G} = 0 holds on the space of antiinvariant class functions. In the following we will therefore assume, that α is *- invariant. Following the argument for the analogous formula [Rog], lemma 12.5.1 we get for all *-invariant class functions α on M

D_{G}(t)α^{G}(t^{δ}) = ε(T) ∑_{w∈W_{F} (T,M)\W_{F} (T,G)} τ(wtw^{−1})κ(w(δ))D_{M}(wtw^{−1})α(wtw^{−1}) .

Here T ⊂ M and t ∈ T^{r}, δ ∈ Ker(H^{1}(F, T) → H^{1}(F, G)). Fix a suitable admissable embedding T → G. We get a homomorphism ρ_{T} of the Galois group of the splitting field of T to the abstract Weyl group W_{G}. Thus the Galois group acts (by conjugation with ρ_{T}) on W_{G}. Let W_{F} (T, G) be the invariants under the Galois group Gal(\overline{F} : F). Finally

let denote W (G, T) = Norm(T, G)(F)/T (F). Similar we define W_{F} (T, M) and W (T, M).

Finally τ is the product over the factors Δ_{I}Δ_{II}Δ_{III} of Langlands and Shelstad ([LS]). The factor ε(T) is one if T^{∗} is conjugate to T in M, otherwise ε(T) = 2. Observe that ε(G)#(W_{F} (T, M) \ W_{F} (T, G)) = 2 for all tori T in M. For further use let us consider the values of α^{G}(t) on the different tori of G and their F-conjugacy classes. These are the following

1) T (F) is isomorphic to x ∈ E, Norm_{E/E^{+}}(x) ∈ F^{+}, where E/F is an extension of F of degree four, whose splitting field is either cyclic of degree four or dihedral of degree eight over F. E contains a quadratic extension field E^{+} of F. Such T do not come from tori in M and therefore α^{G} vanishes on such T.

2) T ≅ {x = (x_{1}, x_{2}) ∈ L_{1}^{∗}×L_{2}^{∗} | Norm_{L_{1}/F} (x_{1}) = Norm_{L_{2}/F} (x_{2})} for two nonisomorphic

quadratic field extensions L_{1}/F and L_{2}/F. Such tori do not come from tori in M and α^{G} vanishes on such tori T.

3) T ≅ L^{∗} × F^{∗} ⊂ Q. Such tori do not come from M and α^{G} vanishes on them.

4) T ≅ {x = (x_{1}, x_{2}) ∈ L^{∗} × L^{∗} | Norm_{L/F} (x_{1}) = Norm_{L/F} (x_{2})} for a quadratic

field extensions L/F. These tori (elliptic case I, see [W1]) come from the tori

L^{∗} × L^{∗}/F^{∗} of M. We have D_{G}(T) = H^{1}(F, T) ≅ ZZ/2ZZ and ε(T) = 1. The Galois

action is via w_{max}, which is the element of greatest length of W_{G}. The Weyl group is the semidirect product of ⟨w_{1}, w_{2}⟩ and the reflection ι attached to the short root. We have w_{max} = w_{1}w_{2} and ιw_{1} = w_{2}ι. Furthermore W_{F} (T, G) = W_{G} and W_{F} (T, M) = W_{M}. The embedding W_{F} (T, M) ↪ W_{F} (T, G) is an isomorphism onto the subgroup generated by the permutation ι and the element w_{max}. The outer automorphism ∗ action on T corresponds to the action of w_{2} ∈ W_{F} (T, G). We

have W (T, M) = W_{F} (T, M). Finally we have W (G, T) = W_{F} (G, T) iff χ_{L/F} (−1) = 1,

in which case T^{δ} is not F-conjugate to T for the nontrivial element δ ∈ D_{G}(T).

In the case χ_{L/F} (−1) = −1 we have T^{δ} ≅ T and W (G, T) = W_{F} (T, M) ⊂ W_{F} (T, G).

5) These tori are T ≅ (L_{1} × L_{2})/F^{∗} for two nonisomorphic quadratic field extensions L_{1}/F and L_{2}/F (elliptic case II, see [W2]). These tori come from M with ε(T) = 2. Then W_{F} (T, G) is the subgroup of W_{G} generated by the two element w_{max}, ι (image of Galois), which can be identified with the image of W_{F} (T, M) = W_{M}.

Also D_{G}(T) ≅ ZZ/2ZZ.

6) These tori are T ≅ L^{∗} × F^{∗} ⊂ P, contained in the Siegel parabolic P for quadratic field extensions L/F. They come from the tori T_{M} = (L^{∗} × (F^{∗})^{2})/F^{∗} of M via

the embedding (x, diag(t_{2}, t′_{2}))/(t · id, t^{−1} · id) ↦ blockdiag(t′_{2}det(x)x^{−t}, t_{2}x). Here x ∈ L^{∗} ⊂ Gl(2, F) and t_{2}, t′_{2} ∈ F^{∗}. We have ε(T) = 2 and D_{G}(T) = 1. Furthermore W_{F} (T, G) is generated by w_{max}, ι and W_{F} (T, G) = W_{F} (T, M) = W_{M}.

7) The split torus T ≅ (F^{∗})^{3}. It comes from M with ε(T) = 1. Furthermore

W (T, G) = W_{F} (T, G) = W_{G} and W (T, M) = W_{F} (T, M) = W_{M}, which maps to

the subgroup ⟨ι, w_{max}⟩ of W_{G}. The admissable embedding is the one used in 6) for diagonal matrices x = diag(t_{1}, t′_{1}). The involution * acts on the diagonal split torus T in M. This action corresponds to the action of the element w_{2} ∈ W_{G}.

Let denote ν_{M}(g_{1}, g_{2}) = det(g_{1}g_{2}) for (g_{1}, g_{2})/(t id, t^{−1} id) ∈ M and let denote ν_{G} the sym-

plectic similitude factor ν_{G} : G → F^{∗}. Then in the cases 4)-7) an admissable isomorphism of tori ψ : T_{M} ≅ T_{G} ⊂ G satisfies ν_{G}(ψ(t)) = ν_{M}(t). Furthermore ν_{G}(t^{δ}) = ν_{G}(t) holds for δ ∈ D_{G}(T). This is easy to verify. It implies (α(t)χ(ν_{M}(t)))^{G} = α^{G}(t)χ(ν(t)) and therefore

\underline{Lemma 1}: χ_{σ_{v}} = ∑ n(π_{v})χ_{π_{v}} implies χ_{σ_{v}⊗χ_{v}} = ∑ n(π_{v})χ_{π_{v}⊗χ_{v}}.

This being said, let us come back to the claim made above. All coefficients of the formula of α^{G} are in Q except those involving α = χ_{σ_{v}}, which has values in E. Therefore

\underline{Lemma 2}: χ_{σ_{v}^{τ}}^{G} = (χ_{σ_{v}}^{G})^{τ}.

By the uniqueness of character relations on gets

n(π_{v}) = n_{τ} (π_{v}^{τ})

where

χ_{σ_{v}^{τ}}^{G} = ∑ n_{τ} (π_{S})χ_{π_{S}} .

\underline{Proof and explanation}: π_{v}^{τ} is defined as follows. Choose any automorphism τ of C which induces τ ∈ Σ_{E}. Choose an admissable basis (extending the basis of suitable fixed spaces of a cofinal system of compact open groups) of the representations space and consider the Q-vectorpace generated by it. Then τ acts on the C-vectorspace preserving the fixed spaces of a cofinite system of compact open subgroups of the representation. Put π_{v}^{τ} (g) = τ ∘ π_{v}(g) ∘ τ^{−1}. This is a C-linear representation of G_{v} on the representation space W of π_{v}. For K in the cofinal system we have, that the subspace W^{K} is stable under τ. Hence W^{π_{v}^{τ} (K)} = τ(W^{π_{v}(K)}) = W^{π_{v}(K)}. Hence π_{v}^{τ} is admissable. Its character satisfies χ_{π_{v}^{τ}} = χ_{π_{v}}^{τ}. The isomorphism class of π^{τ} is independent of the choice of the admissable basis. However it is not clear whether π_{v}^{τ} is preunitary if π_{v} was preunitary. Therefore we can not immediately apply linear independence of characters to deduce the formula wanted. We argue as follows. There are only finitely many π_{S} with n_{τ} (π_{S}) ≠ 0 and nontrivial invariant spaces under a fixed compact open subgroup K_{S}. Denote this set by R_{K_{S}}. For π_{S} ∈ R_{K_{S}} also π_{S}^{τ} has nontrivial invariants under K_{S} (provided it was in the chosen cofinal system, what we will assume). If π_{S}^{τ} ∉ R_{K_{S}}, then there exist a compact open subgroup K such that (π_{S}^{τ})^{K} is not isomorphic to π^{K}, π ∈ R_{K_{S}} as a module under the K-Hecke algebra. This contradicts linear independency of a finite number of irreducible characters of the Hecke algebra. Hence π_{S}^{τ} is in R_{K_{S}} and especially it is unitary. We have

χ_{σ_{v}^{τ}}^{G} = ∑ n_{τ} (π_{S})χ_{π_{S}}
χ_{σ_{v}^{τ}}^{G} = (χ_{σ_{v}}^{G})^{τ} = ∑ n(π_{S})χ_{π_{S}}^{τ}

which is the same as

∑ n(π_{S})χ_{π_{S}^{τ}} .

We therefore get n_{τ} (π_{S}^{τ}) = n(π_{S}).

Of course we also have for the weak lifts of σ^{τ} the corresponding τ-twisted version of our construction above. It gives a decomposition of the (π^{S})^{τ} isotypic component into Galois representations ρ_{τ,π_{S}^{τ}} attached to the (π_{S}π^{S})^{τ}. We have corresponding decompositions (**) in the τ-twisted setting

ρ_{τ,π_{fin}^{τ}} ⊗ ν_{l}^{r} ≅ m_{τ,1}(π_{S}^{τ})ρ_{1}^{τ} ⨁ m_{τ,2}(π_{S}^{τ})(ρ_{2}^{τ} ⊗ ν_{l}^{k_{2}−2}) atached to σ^{τ} .

This of course holds because ρ_{i}^{τ} is associated to σ_{i}^{τ}. Observe that ν_{l} has values in Q_{l}^{∗}.

\underline{Corollary 4}: The set of representations π_{S} with the property n(π_{S}) ≠ 0 is Galois stable in the sense of n(π_{fin}) ≠ 0 ⇔ n_{τ} (π_{fin}^{τ}) ≠ 0 with finite orbits. Furthermore we have

m_{τ,1}(π_{fin}^{τ}) = m_{1}(π_{fin}) and m_{τ,2}(π_{fin}^{τ}) = m_{2}(π_{fin}).

\underline{Proof}: This is clear from what was explained above, provided that m_{τ,1}(π_{fin}^{τ})+m_{τ,2}(π_{fin}^{τ}) = \frac{1}{2}dimρ_{π_{fin}^{τ}} equals \frac{1}{2}dimρ_{π_{fin}}. This follows if we imitate the conjugation by τ in the l-adic setting. This even shows ρ_{π_{fin}^{τ}} ≅ ρ_{π_{fin}}^{τ}. Therefore

ρ_{π_{fin}}^{τ} ⊗ ν_{l}^{r} ≅ m_{1}(π_{fin})ρ_{1}^{τ} ⨁ m_{2}(π_{fin})(ρ_{2} ⊗ ν_{l}^{k_{2}−2})^{τ}
ρ_{π_{fin}^{τ}} ⊗ ν_{l}^{r} ≅ m_{τ,1}(π_{fin}^{τ})ρ_{1}^{τ} ⨁ m_{τ,2}(π_{fin}^{τ})(ρ_{2} ⊗ ν_{l}^{k_{2}−2})^{τ} .

This implies m_{1}(π_{fin}) = m_{τ,1}(π_{fin}^{τ}).

\underline{Corollary 5}: In the situation of corollary 2 we have and

m_{1}(π_{fin}) = dimC(H^{k_{1}+k_{2}−3,0}(S(G), EC(r))(π_{fin}))

m_{2}(π_{fin}) = dimC(H^{k_{1}−1,k_{2}−2}(S(G), EC(r))(π_{fin})) .

\underline{Proof}: Look at the action of the Galois group Gal(\overline{Q_{l}} : Q_{l}). The representations ρ_{1} and ρ_{2} are scalar extensions of E_{λ}-adic representations ρ_{λ,i}. Restriction of scalars defines Q_{l}-adic representations. Observe that

⨁_{τ∈Σ_{E}} ρ_{λ,π_{S}}^{τ}

descends to an l-adic representation of Gal(\overline{Q_{l}} : Q_{l}). It is a subrepresentation of several copies of the l-adic representations

⨁_{τ∈σ_{E}} ρ_{λ,1}^{τ}

and

⨁_{τ∈σ_{E}} ρ_{λ,2}^{τ} ⊗ ν_{l}^{k_{2}−2} ,

which are of Hodge-Tate type. The same is therefore true for the constituent ρ_{λ,π_{S}}.

In the l-adic cohomology H_{!}^{3}(S, E) consider an irreducible l-adic constituent under Gal(\overline{Q} : Q) × G(A A_{fin}). It endomorphism field is a numberfield E. Its extension of scalar to \overline{Q_{l}} is the representation above. It is of Hodge-Tate type.

According to Chai and Faltings [CF], thm 6.2 the weights can be read off from the de Rham cohomology. This remains true for the non projective smooth varieties S_{K}(G). The comparison yields the result above, if we take into account corollary 3 and 4. Thus the corollary 5 is proved.

For the rest of this section let us discuss the case, where π is neither CAP nor a lift from H. The situation is similar to the one above, but simpler.

Let us consider π as in the theorem of [W], section 8 and let us assume, that π is not a lift from M. Write π = π_{p} ⊗ π^{p}, where p temporarily denoted some fixed chosen spherical place. Assume that there is another cuspidal automorphic representation π′ of GSp(4, A A), such that π′ = π′_{p} ⊗ π^{p}. So outside p both representations are assumed to be isomorphic. If π′_{p} is also spherical, then it also contributes to the formula from corollary 1

(∗) 4 · T race_{s}(F rob_{p}^{n}; H_{!,et}(S_{K}(\overline{F}_{p}), E_{λ})(π^{p,∞})) = ∑_{π′} m(π′)tr π′_{p}(h_{p}^{(n)}) .

The sum runs over all irreducible automorphic representations π′ counted with some multiplicities m(π′), such that π′_{v} ≅ π_{v} for v ≠ p, ∞ (and π′_{p} is unramified). Here m(π′) is the multiplicity of π′_{p} in H_{!}(S_{K}(C), E_{λ})(π^{p,∞}). See [W], section 9.

If we now vary p, we see that for almost all p the previously given representation π is uniquely determined by π^{p}. This is clear from the finiteness of the number of automorphic representation for given type at infinity and fixed level. So for almost all primes p the right side of the formula (*) is just the single term m(π)trπ_{p}(h_{p}^{(n)}).

5.Orthogonality Relations

In this section we come back to the multiplicity formula for M-lifts. These formulas are encoded in character identities and involve purely local information. Now let us study these identities for a fixed nonarchimedian place v ∈ S. In the following let F denote the corresponding local field. Every discrete series representation σ_{v}, v ∈ S of M_{v} can be embedded in a global cuspidal representation σ of given discrete nonarchimedian type, unramified outside S. Therefore the above local character expression holds especially for all discrete series representations σ_{v} of M_{v}. Let us therefore put G = G_{v}, M = M_{v}. All representations to be considered will be local ones.

Consider scalar products on the set of elliptic regular elements as in Rogawski [Rog] or Kazhdan [Ka]. For the moment let us fix some central characters. Let α = χ_{σ_{v}} be a ∗-invariant class function on M. We do not assume σ_{v} to be irreducible at the moment, but to be of the form σ_{v} = σ_{v,1} ⊗ σ_{v,1} resp. σ_{v} = (σ_{v,1} ⊗ σ_{v,2}) ⨁(σ_{v,2} ⊗ σ_{v,1}) for irreducible discrete series representations of σ_{v,1}, σ_{v,2} of Gl(2, F) with equal central characters. The L^{2}-norm on the elliptic locus is then

⟨χ_{σ}, χ_{σ}⟩_{M,e} = 1 resp. 2

by the orthogonality relations (see [Ka]). Similarily for irreducible discrete series representations π of G we have

⟨χ_{π}, χ_{π}⟩_{G,e} = 1 .

See also [Rog], prop 12.5.2.

Let α be a ∗-invariant class function on M. Then similar to [Rog], p.183 ff one can show in our case

⟨α_{1}^{G}, α_{2}^{G}⟩_{G,e} = 4⟨α_{1}, α_{2}⟩_{M,e} .

One needs two ingredients for this result:

1) |τ(t)|^{2} = 1 where τ = Δ_{I}Δ_{II}Δ_{III}, which can be directly checked (without much explicit computation)

2) [D_{G}(T) : D_{M}(T)] = 2 for all elliptic tori in M.

Let me repeat the calculation of α^{G}: Let t ∈ T, where T ↪ M is of type 4) above (elliptic case I). Then either χ_{L/F} (−1) = 1. There is one F-conjugation class T (T^{∗} = T) of type 4) in M and two stably conjugate, but not F-conjugate tori T^{δ}, δ ∈ D_{G}(T) in G. Call them T and T′. The map H^{1}(F, T) → H^{1}(F, N(T)) is injective. Its ”kernel” is W_{F} (T, G)/W (T, G) = 1. We get from the Weyl integration formula

\frac{1}{4} ∫_{T/Z} D_{M}(t)^{2}α(t)O_{t}^{M}(f^{M})dt = \frac{1}{8} ∫_{T/Z} D_{G}(t)^{2}α^{G}(t)O_{t}^{G}(f)dt + \frac{1}{8} ∫_{T′/Z} D_{G}(t′)^{2}α^{G}(t′)O_{t′}^{G}(f)dt′ .

The test functions f allow to separate the W (G, T) = W_{G} = ⟨W_{M}, ∗⟩ orbits in T and allows to separate T and T′. Substituting O_{t}^{M}(f^{M}) = (τD_{G}/D_{M})(t)(O_{t}^{G}(f) − O_{t′}^{G}(f)) gives

\frac{1}{4} ∫_{T/Z} D_{M}(t)D_{G}(t)τ(t)α(t)(O_{t}^{G}(f) − O_{t′}^{G}(f))dt = \frac{1}{8} ∫_{T/Z} D_{G}(t)^{2}α^{G}(t)O_{t}^{G}(f)dt

+\frac{1}{8} ∫_{T′/Z} D_{G}(t′)^{2}α^{G}(t′)O_{t′}^{G}(f)dt′ .

After making the left side W_{G} invariant, we get

D_{G}(t)α^{G}(t) = τ(t)D_{M}(t)α(t) + τ(t^{∗})D_{M}(t^{∗})α(t^{∗})
= τ(t)D_{M}(t)(α(t) + α(t^{∗})) .

and

α^{G}(t^{δ}) = κ(δ)α^{G}(t) .

Observe τ(t^{∗}) = χ_{L/F} (−1)τ(t) = τ(t).

Next consider the tori T of type 4) in the case χ_{L/F} (−1) = −1. Then there is a unique F-conjugation class of T in M and in G. The map H^{1}(F, F) = D_{G}(T) → H^{1}(F, N(T)) is trivial and D_{G}(T) = W_{F} (T, G)/W (T, G) (via the exact sequence attached to W_{G} = N(T)/T

over \overline{F}). We have W (T, G) = W_{F} (M, T) = W_{M} and W_{F} (T, G) = W_{G} = ⟨W_{M}, ∗⟩. We get

from the Weyl integration formula

\frac{1}{4} ∫_{T/Z} D_{M}(t)^{2}α(t)O_{t}^{M}(f^{M})dt = \frac{1}{4} ∫_{T/Z} D_{G}(t)^{2}α^{G}(t)O_{t}^{G}(f)dt .

The test function allow to separate the W (T, G) orbits in T. We get with t′ = t^{∗}

\frac{1}{4} ∫_{T/Z} D_{M}(t)D_{G}(t)τ(t)α(t)(O_{t}^{G}(f) − O_{t′}^{G}(f))dt = \frac{1}{4} ∫_{T/Z} D_{G}(t)^{2}α^{G}(t)O_{t}^{G}(f)dt ,

thus

D_{G}(t)α^{G}(t) = τ(t)D_{M}(t)α(t) − τ(t^{∗})D_{M}(t^{∗})α(t^{∗}) = τ(t)D_{M}(t)(α(t) + α(t^{∗})) ,

and α^{G}(t^{δ}) = κ(δ)α(t) because of τ(t^{∗}) = χ_{L/F} (−1)τ(t) = −τ(t).

Finally consider the case of tori of type 5) (elliptic case II). There are two F-conjugation classes T and T^{∗} of such tori in M. The embeddings of T and T^{∗} into G in [W2] give the same Δ-factor. Observe, that t^{∗} = (t_{1}, t_{2})^{∗} = (t_{2}, t_{1}) and Δ depends only on t_{1}t_{2}. Thus the argument of [Rog], p. 183-184 gives (with an additional contribution from T^{∗} in M!)

D_{G}(t)α^{G}(t^{δ}) = ∑_{w∈W_{F} (T,M)\W_{F} (T,G)} τ(t^{w})D_{H}(t^{w})κ(w(δ))(α + α^{∗})(t^{w}) .

Then W_{F} (T, M) = W_{F} (T, G) implies

D_{G}(t)α^{G}(t^{δ}) = κ(δ)τ(t)D_{M}(t)(α + α^{∗})(t) .

Now we compute for α = α^{∗} resp. α + α^{∗} = 2α

⟨α^{G}, α^{G}⟩_{G,e} = ∑_{T ell.⊂G/≃} #W_{F} (T, G)^{−1} ∫_{T/Z} |τ(t)|^{2}D_{M}(t)^{2}|2α(t)|^{2}

= ∑_{T ell.⊂G/≃st.} #D_{G}(T)#W_{F} (T, G)^{−1} ∫_{T/Z} |τ(t)|^{2}D_{M}(t)^{2}|2α(t)|^{2}

= ∑_{T ell.⊂M/≃st.} #D_{G}(T)#W_{F} (T, G)^{−1} ∫_{T/Z} |τ(t)|^{2}D_{M}(t)^{2}|2α(t)|^{2} .

The sum is over all elliptic tori T in G, which come from M, modulo conjugation resp. stable conjugation.

Each of elliptic tori types of case II gives the following contributions: #D_{G}(T) = 2 and W_{F} (T, G) = W_{F} (T, M) = W (T, M) imply, that the case II contribution is equal to the sum over the two tori T, T^{∗} in M

4 ∑_{T′=T,T^{∗}} #W (T′, M)^{−1} ∫_{T′/Z} D_{M}(t)^{2}|α(t)|^{2}

by our assumption α = α^{∗}. The elliptic tori of case I give the remaining contributions:

For χ_{L/F} (−1) = 1

\frac{1}{8}(∫ D_{M}(t)^{2}|τ(t)|^{2}|2α(t)|^{2}dt + ∫ D_{M}(t)^{2}|τ(t)|^{2}|2α(t)|^{2}dt)

and for χ_{L/F} (−1) = −1

\frac{1}{4} ∫ D_{M}(t)^{2}|τ(t)|^{2}|2α(t)|^{2}dt ,

hence in both cases

\frac{1}{4} ∫_{T/Z} D_{M}(t)^{2}|2α(t)|^{2}dt .

There is a single F-conjugation class of T in M and #W (T, M) = 4, hence this is

4 ∑_{T′=T} #W (T′, M) ∫_{T′/Z} D_{M}(t)^{2}|α(t)|^{2}dt .

Adding everything together therefore proves for α = α^{∗} the formula claimed

\underline{Lemma 3}: ⟨α^{G}, α^{G}⟩_{G,e} = 4⟨α, α⟩_{M,e}.

Now consider irreducible admissable representations σ_{v} = σ_{v,1} ⊗ σ_{v,2} of M of the discrete series. We distinguish two cases: The first case where σ_{v}^{∗} is not isomorph to σ_{v}, the second where σ_{v}^{∗} ≅ σ_{v}. The second case takes place iff σ_{v,1} is isomorphic to σ_{v,2}. The orthogonality relations for M give ⟨χ_{σ_{v}+σ_{v}^{∗}}, χ_{σ_{v}+σ_{v}^{∗}}⟩_{M,e} = 2 in the first case. Because of χ_{σ_{v}^{∗}}^{G} = χ_{σ_{v}}^{G} we therefore get

\underline{Lemma 4}: We have

⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{G,e} = 2

in the first case (σ is not ∗ invariant) and

⟨χ_{σ}^{G}, χ_{σ}^{G}⟩_{G,e} = 4

in the second case (σ_{v} is ∗-invariant).

The pairing ⟨., .⟩_{∗,e} annihilates the spaces R_{I}(G) resp. R_{I}(M) generated by characters of fully induced representations.

One can rearrange the character expression in the form

χ_{σ_{v}}^{G} ∈ ∑ ñ(π_{v})χ_{π_{v}} + ∑_{P ≠G} ∑_{ρ_{v}} Ind_{P}^{G}(σ_{v})

such that only tempered representations π_{v} occur in the first summation with integer coefficients ñ(π_{v}). For that see [Ka], p.6 or more precisely [B-W] chapter XI. 2.11 and 2.13. Namely every nontempered representation π_{v} occurs as a Langlands quotient J(P, σ_{v}, χ_{v}) of a standard induced representation I(P, σ_{,}χ_{v}). The multiplicity d(π_{v}) of the Langlands quotient π_{v} ≅ J(P, σ_{v}, χ_{v}) in the composition series of I(P, σ_{v}, χ_{v}) is one [B-W], XI, 2.13. Therefore π_{v} can be expressed in the integral Grothendieck group by the class of the induced representation I(P, σ_{v}, χ_{v}) in R_{I}(G) and some integral linear combination of nontempered or tempered representations with smaller Langlands parameter (again [B-W], XI 2.13). All the nontempered representations occuring in this expression can again be written as Langlands quotients. This can be continued. By the finiteness of exponents of a given (induced) representation, this process terminates after finitely many steps. This allows to express every nontempered representation as a ZZ-linear combination ∑ ñ(π_{v})π_{v} + R_{I}(G) of tempered ones modulo R_{I}(G). Substituting into the summation gives the expansion χ_{σ_{v}}^{G} = ∑ ñ(π_{v})χ_{π_{v}} with integral coefficients ñ(π_{v}) valid modulo a sum with summands in R_{I}(G).

However, one general remark is in order. It remains the question, whether the two new sums are still absolutely convergent. This is true. Let us write our character sum in the form indexed by infinitesimal characters χ. This means the following. χ is a class of pairs (L, ρ_{v}), where L is a standard Levi subgroup and ρ_{v} is cuspidal irreducible representation of L, under conjugation in G. Fix some parabolic P = LN belonging to a representative (L, ρ_{v}) of the infinitesimal character and consider all representations occuring as constituents of representations induced from cuspidal representations of the Levi component of P. Call the finite set of inequivalent representatives R_{χ}. Write

χ_{σ_{v}}^{G} = ∑_{χ} π_{χ}

π_{χ} = ∑ n_{χ,i}χ_{π_{i}} , π_{i} ∈ R_{χ} .

Choose a cofinal system of good small open compact subgroups K of G_{v} in the sense of [Be]. We say, the an infinite sum ∑ n_{χ}π_{χ} is locally finite, if for every K in the cofinal system there exists only a finite number of χ, such that χ = (L, ρ_{v}), ρ_{v}^{K∩L} ≠ 0 and π_{χ} ≠ 0. Let f be a K-biinvariant function. Then π_{χ}(f) = 0 unless σ^{K∩L} ≠ 0. Therefore locally finite implies absolute convergent. Conversely every sum ∑ n(π_{v})χ_{π_{v}}, such that for every K there only finitely many π_{v} with π_{v}^{K} ≠ 0 and n(π_{v}) ≠ 0, is locally finite. This follows, from [Be], prop. 3.8. Namely π_{v} ∈ R_{χ} and π_{v}^{K} = 0 implies ρ_{v}^{L∩K} = 0. Applying Jaquets functor is compatible with infinitesimal characters, therefore preserves locally finite sums.

It is clear this, that our expression for χ_{σ_{v}}^{G} is locally finite. Our rearrangement made above gives two locally finite sums, the sum ñ(π_{v})π_{v} and the sum of induced representations in R_{I}(G). This is clear, because we only splitted the inner sums π_{χ} = π̃_{χ} + π_{I,χ}, whenever

π_{χ} ≠ 0.

\underline{Method 1}: Let f = ∑_{i=1}^{r} ñ(π_{i})f_{π_{i}} be a finite linear combination of pseudo coefficients f_{π_{i}} of discrete series representations π_{i}. Then f is locally constant with compact support, has vanishing orbital integrals outside the elliptic locus and orbital integrals are given by characters on the elliptic locus. Now χ_{σ_{v}}^{G} = ∑ ñ(π_{v})χ_{π_{v}} + ∑_{P ≠G,ρ_{v}} Ind_{P}^{G}(ρ_{v}) where the first is a locally finite sum of tempered characters. Characters of induced representations vanish on the elliptic locus. Therefore we get

∫_{G} f(g)χ_{σ_{v}}^{G}dμ = ∫_{G^{e}} f(g)χ_{σ_{v}}^{G}dμ = ∑ ñ(π_{v})χ_{π_{v}}(f) = ∑_{i=1}^{r} ñ(π_{v})^{2} ,

because χ_{π′}(f_{π}) = 0 holds for all tempered irreducible π′ not isomorphic to π and χ_{π}(f_{π}) = 1. On the other hand, the Schwartz inequality gives

| ∫_{G^{e}} f(g)χ_{σ_{v}}^{G}dμ|^{2} ≤ ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{G,e}⟨f, f⟩_{G,e} = ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{G,e} ∑_{i=1}^{r} ñ(π_{i})^{2} .

Equality holds iff χ_{σ_{v}}^{G} = O^{G}(f) on the elliptic locus G^{e}. Using the formula above we can bring this inequality into the simpler form

∑_{π_{v}∈Π^{2}(G_{v})} ñ(π_{v})^{2} ≤ ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{e} ,

where the sum extends over all isomorphism classes Π^{2}(G) of discrete series representations, letting r tend to infinity.

Now we use that the integrality condition ñ(π_{v}) ∈ ZZ holds. On the other hand, for irreducible σ_{v} the value of ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{e} is either two or four. Therefore

ñ(π_{v}) = ±1 or 0

holds for discrete series representations π_{v} (and all globally realizable σ_{v}, especially those σ_{v} which are square integrable); furthermore ñ(π_{v}) ≠ 0 can happen for at most two of them (if σ_{v} is not ∗-invariant) and for at most four of them (if σ_{v} is ∗-invariant). The case ñ(π_{v}) = ±2 for discrete series π_{v} cannot occur! This is explained in the next paragraph.

If there are two resp. four nontrivial discrete series contributions with coefficient ±1 or one with coefficient ±2, then this determines χ_{σ_{v}}^{G} on the elliptic locus. If not, then there exist limits of discrete series which compensate the difference on the elliptic set. Tempered irreducible representations are called limits of discrete series, provided they are nontrivial on the elliptic locus. This excludes for example all tempered representations, which are fully induced.

Assume the discrete series show up with coefficients ñ(π_{v}), such that χ_{σ_{v}}^{G} = ∑_{i=1}^{r} ñ(π_{i})χ_{π_{i}} holds on the elliptic locus. Now χ_{σ_{v}}^{G} vanishes identically on elliptic tori of G corresponding to a cyclic field extension of Q_{v} of degree four. Following Rogawski [Rog] p.194-195 any finite character relation ∑_{i=1}^{r} ñ(π_{i})χ_{π_{i}} = 0 between characters of the discrete series on this elliptic torus implies ∑ d(π_{v})ñ(π_{v}) = 0. As formal degrees d(π_{v}) are positive, not all the signs of the ñ(π_{i}) can be positive respectively negative. This especially excludes the case r = 1 and therefore the possibility ñ(π_{v}) = ±2 for discrete series π_{v}, as mentioned above. It furthermore implies in the case of representations σ_{v}, which are not ∗-invariant, that among the two possible numbers ñ(π_{1}), ñ(π_{2}) = ±1 for discrete series representation π_{v} under consideration there is exactly one of them 1 and one of them −1. Let us denote the two irreducible discrete series representations, for which ñ(π_{v})^{2} = 1 holds, by π_{v}^{+} and π_{v}^{−}. This can be done in such a way that

χ_{σ_{v}}^{G} = χ_{π_{v}^{+}} − χ_{π_{v}^{−}}

holds on the elliptic locus. Or otherwise at most one discrete series representations π_{v} has a coefficient ñ(π_{v}) ≠ 0.

\underline{Method 2}: Assume, that \underline{asymptotic} independence holds for a certain infinitesimal character χ = (L, ρ_{v}), P_{L} = LN of the Bernstein center. This means the following. Let π = ∑ n_{i}π_{i}, n_{i} ∈ ZZ be a finite virtual representation defined from representations of the class R_{χ}. The Jaquet module π(P_{L}) of π is a virtual cuspidal representation of the Levi component L. Let AD_{χ} be the finitely generated ZZ-module of all ∑ n_{i}π_{i}, such that π_{i} ∈ R_{χ} and such that ∑ n_{i}π_{i}(P_{L}) = 0 in R(L). We say, that asymptotic independence holds for the character χ, if AD_{χ} = 0. For instance one has AD_{χ} = 0 for χ = (L, ρ), unless the pair is ”ramified”, which means that ρ is nontrivially stabilized by the Weyl group of a parabolic attached to L. Otherwise every generalized exponent in the Jaquet module of the induced representation Ind_{P_{L}}^{G}(ρ_{v}) with respect to P_{L} has multiplicity one, which then easily implies asymptotic independence.

Next suppose that f is locally finite sum f = ∑_{χ} π_{χ} where each π_{χ} is a (finite) linear combination π_{χ} = ∑ n_{χ,i}χ_{π_{χ,i}} ∈ R_{χ} of characters of irreducible admissable representations π_{χ,i} with infinitesimal character χ. To such a sum f and a parabolic subgroup Q we associate the locally finite sum

f(Q) = ∑_{χ} π_{χ}(Q) ,

where each π_{χ}(Q) = ∑ n_{χ,i}π_{χ,i}(Q) ∈ R(L_{Q}) is the sum of the characters of the Jaquet module of the representations π_{χ,i} with respect to Q.

Suppose f(Q) = 0 holds. Claim: Then π_{χ}(Q) = 0 holds for each single infinitesimal character χ = (L, ρ_{v}) of G. Proof: Let M be the Levi component of Q. By the ”shuffle” formula of [DBK], p.189 one has r_{MG}∘i_{GL} = ∑_{w∈W (M)\W/W (L)} i_{M,wLw^{−1}∩M} ∘w∘r_{L∩w^{−1}Mw,L} for minimal lenght representatives w. If ρ_{v} is a cuspidal irreducible representation of L, then r_{MG}i_{GL}(ρ_{v}) vanishes unless L ⊂ w^{−1}Mw for some w. Then r_{MG} ∘ i_{GL} = ∑_{w∈W (M)\W/W (L),wLw^{−1}⊂M} i_{M,wLw^{−1}}w(ρ_{v}). Therefore r_{MG}(π_{v}) is, for all irreducible π_{v} with infinitesimal character (L, ρ_{v}), a linear combination of representations in M with infinitesimal characters w(L, ρ_{v})w^{−1}, wLw^{−1} ⊂ M. Hence, if the infinitesimal characters of π_{v} and π′_{v} are different for G, the infinitesimal characters of the constituents of r_{MG}(π_{v}) and r_{MG}(π′_{v}) are different for M. Especially r_{MG}(∑_{χ} π_{χ}) = 0 implies r_{MG}(π_{χ}) = 0 for all χ. But r_{MG} is up to a normalisation just the virtuel Jaquet module, so π_{χ}(Q) = 0 follows for all χ. It should be remarked, that we implicitely used [Be], §3 in the preceeding arguments.

\underline{Remark}: π_{χ}(Q) = 0 implies, by the inductivity of the Jaquet functor, π_{χ} ∈ AD_{χ} for all infinitesimal characters (L, ρ_{v}), such that Q contains some parabolic group P_{L} = LN.

Now recall the theorem of Deligne-Casselman [Ca]. Let G be a reductive group over a p-adic local field. Choose a minimal parabolic subgroup P_{∅} with simple roots Δ. For a subset θ ⊂ Δ on defines the torus A_{θ} = ∩_{α∈θ}kern(α) and P_{θ} = Zentr(A_{θ})R_{u}(P_{θ}). The unipotent radical being generated by the unipotent subgroups belonging to the positive roots, which are not linear combinations of roots in θ. There exists a reductive group L_{θ}, such that Zentr(A_{θ}) = L_{θ} holds. The roots in Δ \ θ define characters on A_{θ} ⊂ L_{θ}. An element a ∈ A_{θ} is called P_{θ}-contractive, if

|α(a)| < 1 , for all α ∉ θ .

Let A_{θ}^{−} denote the set of P_{θ} contractive elements.

Let g be a regular semisimple element of G Let A be the maximal split torus in Z_{G}(g). Some power of g has the form g = a · s, a ∈ A, s ∈ S, where S is the maximal anisotropic torus in Z_{G}(g). After replacing g by a suitable conjugate, there exists a standard parabolic group P_{θ}, such that a ∈ A_{θ} = ∩_{α∈θ}kern(α) holds and

|α(a)| ≤ 1 for all α ∈ Δ .

Δ is the set of simple roots associated to the standard minimal parabolic group P_{∅}. Let θ ⊂ θ(g) ⊂ Δ be the set of simple roots with the property |α(a)| = 1. Let denote P_{θ} ⊂ P ⊂ G denote the parabolic attached to θ(g). The conjugacy class of g is said to belong to the P- stratum. If g is semisimple regular, then for the character χ_{π_{v}} of a finitely generated admissable representation π_{v} of G

χ_{π_{v}}(g) = χ_{π_{v}(P)}(g)

holds according to [Ca], Thm 5.2 . Similarily one proves for Q = P_{θ}, A_{Q} = A_{θ} and L_{Q} = L_{θ} (for some θ in Δ) the following.

\underline{Lemma 5}: Suppose f is a locally finite sum of characters of a reductive p-adic group G. Suppose that the invariant distribution f is represented by a locally integrable function, on the set G^{rs} of semisimple, regular element of G, also denoted f. In other words f(ϕ) = ∫_{G^{rs}} f(g)ϕ(g)dg holds for all locally constant functions ϕ on G with compact support. Let Q = L_{Q}N_{Q} be a parabolic subgroup of G with Levi component L_{Q} and the split component A_{Q}. Then the following conditions are equivalent

(i) Let Ω ⊂ L_{Q} be a compact subset. For all semisimple regular elements g ∈ Ω one has f(ag) = 0 for all sufficantly Q-contractive elements a ∈ A_{Q}^{−}, i.e |α(a)| < ϵ(Ω) << 1 for all roots α ∉ θ.

(i)’ For all semisimple regular elements g ∈ Ω one has f(ag) = 0 for all elements a of A_{Q}^{−} with |α(a)| < ϵ(Ω) << 1 for α ∉ θ, for which (log(|α(a)|))_{α∉θ} is contained in a fixed open cone of the Weyl chamber attached to A_{Q}^{−}.

(ii) f(Q) = 0 holds in the sense above.

\underline{Proof}: Let Q^{−} = L_{Q}N_{Q}^{−} be the parabolic opposite to Q = L_{Q}N_{Q}. Choose some small good compact open subgroup K ⊂ G such that

K = (K ∩ N_{Q}^{−})(K ∩ L_{Q})(K ∩ N_{Q}^{+}) .

Put K̃ = K ∩ L_{Q}.

Let g ∈ L_{Q} be regular semisimple and let a ∈ A_{Q} be Q-contractive. By replacing a by its power and g by a^{n}g one can assume that a, g lie in

L_{Q}^{−} = {g ∈ L_{Q} | g(K ∩ N_{Q})g^{−1} ⊂ K ∩ N_{Q} and g(K ∩ N_{Q}^{−})g^{−1} ⊂ K ∩ N_{Q}^{−}} .

Then we obtain

(∗) f(meas(K_{i}agK_{i})^{−1}1_{K_{i}agK_{i}}) = f(Q)(meas(K̃_{i}agK̃_{i})^{−1}1_{K̃_{i}agK̃_{i}}) .

Observe, that only finitely many irreducible admissable characters intervene on both sides of the equality, thus we get this from [BDK], 5.3 term by term. Remark: If the distributions f and f(Q) are both smooth at the point ag, then by taking the limit K → 0

we obtain f(ag) = f(Q)(ag).

Suppose (ii) holds. Then the identity (*) above shows, that f(meas(K_{i}agK_{i})^{−1}1_{K_{i}agK_{i}}) vanishes for all a ∈ A_{Q}^{−}, which are highly contractive. This is independent from K. By a covering argument this implies ∫_{C} f(g)dg = 0 for all compact subsets C contained in this range of points. This gives f(ag) = 0 (almost everywhere) for a ∈ A_{Q}^{−}, which satisfy |α(a)| < ϵ(g), α ∉ θ. It is clear that the argument work locally uniformly.

Conversely assume f(ag) = 0 for all highly Q-contractive a ∈ A_{Q}^{−}. Assume K to be chosen small enough, such that every element in KagK is conjugate to an element a′g′

with a′, g′ ∈ L_{Q}^{−} and g′ ∈ Ω(g) and a′ satisfying |a′| ≤ ϵ(Ω). Then f(meas(KagK)^{−1}1_{KagK})

vanishes and therefore

f(Q)(meas(K̃_{i}agK̃_{i})^{−1}1_{K̃_{i}agK̃_{i}}) = ∑ n_{i}ω_{σ_{i}}(a)χ_{σ_{i}}(meas(K̃_{i}agK̃_{i})^{−1}1_{K̃_{i}agK̃_{i}})

vanishes. The sum runs over the irreducible representations σ_{i} of L_{Q}, which appear in f(Q). This sum becomes actually a finite sum. Almost all sumands are zero. Using linear indepency for a finite number of different central characters ω_{σ_{i}} on cones as in CAP, lemma 5 we get the following. If this finite sum vanishes for all a in a open subcone of A_{Q}^{−}, then the sum vanishes for all a ∈ A_{Q}. This shows, that for every g ∈ L_{Q} and every

a ∈ A_{Q}) f(Q)(1_{K̃agK̃}) = 0 holds.

How do the K̃ depend on the particular elements a, g ∈ L_{Q} we started from? I claim, that the good small subgroup K = K(a, g) can be chosen such that K̃ = L_{Q} ∩ K(a, g) is a suitably fixed small subgroups K̃ of L_{Q} independent of a and g. This would imply f(Q)(1_{K̃agK̃}) = 0 for all ag ∈ L_{Q}, because the regular semisimple elements are dense. Then f(Q) = 0 follows by letting K̃ get arbitrary small.

To construct K we use the remarks of [Be], p.16. Choose a lattice Λ to construct K as in loc. cit. If we replace Λ by p^{i}Λ we get a cofinal system of good small subgroups K_{i} of G with respect to Q and a^{m} resp a^{n}g for suitable n, m. Now choose a small compact open K_{0} in L_{Q}, which stabilizes Λ ∩ Lie(N_{Q}) and Λ ∩ Lie(N_{Q}^{−}) and Λ ∩ Lie(L_{Q}). Then K_{0} normalizes all K_{i} or equivalently the intersections of K_{i} with N_{Q}, N_{Q}^{−}, L_{Q}. Therefore we have (K_{0}K_{i}) ∩ N_{Q} = K_{i} ∩ N_{Q} and (K_{0}K_{i}) ∩ N_{Q} = K_{i} ∩ N_{Q}, as the decompositions N_{Q}^{−}L_{Q}N_{Q} are unique. Then also we have

(K_{0}K_{i}) = (K_{0}K_{i} ∩ N_{Q}^{−})(K_{0}K_{i} ∩ L_{Q})(K_{0}K_{i} ∩ N_{Q}^{+}) .

Therefore not only K_{i} but also K_{0}K_{i} = K_{i}K_{0} satisfies the assumptions necessary for the argument in [BDK], 5.3. But now ⋃_{i}(K_{0}K_{i}) = K_{0}. Therefore for i sufficantly large K(a, g) can be chosen to be of the form K_{0}K_{i}. Then K̃ = K_{0} for i sufficantly large. This completes the proof of the lemma.

\underline{Example}: If σ_{v} is a cuspidal irreducible representation of M, then it follows that the distribution f = χ_{σ_{v}}^{G} satisfies condition (i) of lemma 5 for all three conjugacy classes of proper parabolic subgroups. Therefore χ_{σ_{v}} has to be a locally finite sum of cuspidal characters and characters with infinitesimal character χ, for which AD_{χ} ≠ 0 holds.

\underline{Method 3}:

Assume σ_{v} ≅ Ind_{B_{M}}^{M} (ρ_{v}) is induced from a character of a Borel subgroup B_{M} of M. Then the character χ_{σ_{v}} has support in the conjugates of the split torus. On the split torus it is given up to the normalisation factor D_{M} by the 4 conjugates of ρ

D_{M}(t)χ_{σ_{v}}(t) = ∑_{w∈W_{M}} ρ_{v}(t^{w}) , t ∈ T_{split}(M) .

According to the formula for α^{G} stated above (see also [Rog], 12.5.1) therefore

D_{G}(t)χ_{σ_{v}}^{G}(t) = ∑_{w∈W_{M} \W_{G}} D_{M}(t^{w})χ_{σ_{v}}(t^{w}) , t ∈ T_{split}(G) ≅ T_{split}(M) .

Observe that in this case κ is trivial and W_{F} (T, M) = W_{M} and W_{F} (T, G) = W_{G} and τ = 1. Fix an admissable isomorphism of split tori of M and G respectively. This includes also a choice of Borelgroups B_{M} resp. B containing the split torus. View ρ_{v} as a character of both split tori via the chosen admissable isomorphism. Because χ_{σ_{v}}^{G} vanishes on the non split conjugacy classes of tori in G, the formulas above imply

(χ_{Ind_{B_{M}}^{M} (ρ_{v})})^{G} = χ_{Ind_{B}^{G}(ρ_{v})} .

The similar formula holds for representations σ_{v} induced from a Levi subgroup of a proper maximal parabolic subgroup P_{M} of M. In that case the lifted character χ_{σ_{v}}^{G} = \frac{1}{2}χ_{σ_{v}+σ_{v}^{∗}}^{G} is the character of a corresponding representation induced from the Levi subgroup of the Siegel parabolic P. Observe, the factor \frac{1}{2} cancels against ε(T) = 2 and that W_{F} (T, M) = W_{F} (T, G) in this case. On nonsplit tori T ≅ L^{∗} × F^{∗} ⊂ P_{M} one of the characters of σ_{v}, σ_{v}^{∗} vanishes. Furthermore the factor Δ_{IV} = D_{G}/D_{M} = D_{P} /D_{P_{M}} compensates the factors appearing in the character formula for induced representations. Finally Δ_{I} = 1 (Hilbert 90), Δ_{II} = 1 (the Galois group stabilizes the set of positive roots outside of M and the corresponding χ-subdata can be chosen trivially). Finally Δ_{III} is a character, which would be enough for our purposes. An explicit calculation gives Δ_{III} = 1. Assume σ_{v} is

induced from the representation ρ_{v}((x, diag(t_{2}, t′_{2}))/(t id, t^{−1}id)) = τ_{v}(x)χ_{1,v}(t_{2})χ_{2,v}(t′_{2}) of a

parabolic subgroup in M with Levi quotient Gl(2, F) × (F^{∗})^{2}/F^{∗}, where (x, diag(t_{2}, t′_{2})) ∈ Gl(2, F) × (F^{∗})^{2} are representatives and where the central character ω_{τ_{v}} of τ_{v} satisfies ω_{τ_{v}} = χ_{1,v}χ_{2,v}. Then we get

(χ_{Ind_{P_{M}}^{M} (ρ_{v})})^{G} = χ_{Ind_{P}^{G}(ρ̃_{v})} ,

where ρ̃_{v} is given by ρ̃_{v}(blockdiag(λg^{−t}, g)) = τ_{v}(g)χ_{2,v}(λ/det(g)) for λ ∈ F^{∗}, g ∈ Gl(2, F).

It remains to check semisimple regular elements, whose centralizer is the split torus.

For that case observe, that the character χ_{Ind_{P}^{G}(ρ̃_{v})} is invariant under w_{2} ∈ W_{G}, which corresponds to the involution * on the split torus of M. This compensates the factor

1/2.

\underline{Corollary 6:} For any irreducible admissable representation σ_{v} of the local group M_{v} the lift χ_{σ_{v}}^{G} can be expressed as a locally finite sum χ_{σ_{v}}^{G} = ∑ n(π_{v})χ_{π_{v}} of irreducible characters of the group G_{v} with integral coefficients n(π_{v}) ∈ ZZ.

\underline{Proof}: It is enough to consider generators of the Grothendieck group. The group M has no elliptic tempered representations, which are not discrete series representations. Hence every irreducible character χ_{σ_{v}} can be written as a finite integral linear combination of discrete series characters and a finite number of induced representations. Discrete series representations σ_{v} are embeddable into global cuspidal representations with fixed archimedian discrete series component. Therefore the global construction already discussed proves the claim in that case. For induced representations the claim follows from the remarks above.

6.Classification of representations

Consider the symplectic group of similitudes GSp(4, F) defined by all matrices g ∈ Gl(4, F) satiesfying g^{t}Jg = λ(g)J, λ(g) ∈ F^{∗}, where

J = ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) .

Our standard parabolic subgroups are supposed to contain the Borel group B = B consisting of all matrices g = ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ GSp(4, F), for which A is upper triangular.

In the first two sections we consider representations induced from cuspidal representations of maximal parabolic subgroups. According to Shahidi [Sh], thm. 6.1 and Waldspurger [Wa], section 5 such representations have a multiplicity one decomposition series. There are two maximal proper parabolic subgroups P and Q up to conjugation.

\underline{The Siegel parabolic P (Results of Shahidi)}:

In that case, the induced representation sometimes contain discrete series representations as shown by [Sh3], prop. 6.1 and thm.5.1. The result of [Sh3] is the following: Let σ_{1} and χ be irreducible representations of Gl(2, Q_{v}) resp. Gl(1, Q_{v}) and let

σ = σ_{1} ⊠ χ

be an irreducible representation of the Levi component of the Siegel parabolic M ≅ Gl(2)× Gl(1). This means σ(x, τ) = σ_{1}(x)χ(τ). Then consider the representation I(σ) = Ind_{P}^{G}(σ) of G induced (with unitary normalization) from

( \begin{matrix}x & ∗ \\ 0 & τ · x^{−t} \end{matrix} ) ↦ σ_{1}(x)χ(τ) .

\underline{Short notation}: I(σ) = σ_{1} ◁ χ.

For cuspidal unitary σ_{1} this induced representation is always irreducible. Note

(σ_{1} ◁ χ) ⊗ χ′ ≅ σ_{1} ◁ χχ′
ω_{σ_{1}◁χ} = ω_{σ_{1}}χ^{2} .

More generally consider the representation σ_{1}ν^{s} ◁ χ, where σ_{1}ν^{s}(x) = σ_{1}(x)|det(x)|^{s} and where σ_{1} is \underline{unitary cuspidal}. In the notation of [Pic] this coincides with Π(σ, s) ⊗ ν^{s}.

Then the representation σ_{1}ν^{s} ◁χ for unitary cuspidal σ_{1} is reducible if and only if s = ±1/2 and furthermore the central character ω_{σ_{1}} of σ_{1} is trivial ω_{σ_{1}} = 1.

In the reducible case s = 1/2 the (Langlands)-quotient J_{P} (ν^{1/2}, σ_{1} ⊠ χ) is irreducible unitary nontempered and not generic. The kernel is a generic discrete series representation D = δ(σ_{1}ν \frac{1}{2} ◁ χ) according to Silberger.

The representation D is irreducible. Because of Bruhat theory ([Si], cor.2.12.2(1)) the induced representation is multiplicity free, as σ_{1}ν^{1/2} is not stable under the nontrivial element in the Weyl group. By the same consideration there are at most two exponents, therefore there are at most two irreducible constituents. Hence D is irreducible.

\underline{The second maximal parabolic Q (Results of Waldspurger)}:

This case was studied by Waldspurger [Wa]. Let σ be χ ⊠ ρ, such that χ and ρ are irreducible cuspidal representations of Gl(1) and Gl(2), combined to a representation of the Levi subgroup Gl(1) × Gl(2). Then the representation I(σ) = Ind_{Q}^{G}(σ) induced (with unitary normalization) from the representation σ

( \begin{matrix}a & 0 & b & ∗ \\ ∗ & x & ∗ & ∗ \\ c & 0 & d & ∗ \\ 0 & 0 & 0 & y \end{matrix} ) ↦ χ(x)ρ(( \begin{matrix}a & b \\ c & d \end{matrix} ))

of Q is multiplicity free and has at most two composition factors.

\underline{Short notation}: I(χ ⊠ ρ) = χ × ρ.

Note

(χ × ρ) ⊗ χ′ = χ × (ρ ⊗ χ)
ω_{χ×ρ} = χω_{ρ} .

The representation χ × ρ is reducible iff one of two conditions

1) χ = 1

2) χ = ν^{±1}χ_{0} and ρ has CM-multiplication by χ_{0}. By this we mean χ_{0}^{2} = 1, χ_{0} ≠ 1

such that ρ ⊗ χ_{0} ≅ ρ.

holds.

Reducibility in the cases 2) yields an irreducible non tempered and an irreducible discrete series representation δ(νχ_{0} × ρ). This follows again from Silbergers result stated in [Sh].

Reducibility in case 1) yields two tempered representations W p_{+}(ρ), W p_{−}(ρ) of G, which are relatives in the sense of Kazhdan, such that the unitary representation decomposes into two irreducible summands

W p_{+}(ρ) ⊕ W p_{−}(ρ) = 1 × ρ .

Of course r_{GP} (W p_{±}(ρ)) = 0 and r_{QG}(W p_{±}(ρ)) = 1 × ρ. Exactly one of the two representations admits a Whittaker model . See also [Wa], p.64 the remark following lemma 8.2 and our more detailed discussion later. Let W p_{+}(ρ) denote the representation with the Whittaker model. We will extend the definition of W p_{+}(ρ) and W p_{−}(ρ) to the case of all discrete series representations ρ of Gl(2, F). See the discussion of case b) below.

\underline{The Borel group B (Results of Tadic, Rodier)}:

In this section we give a review of results of Tadic [T] on the (nonunitary) principal series representations. They are induced representations from the representation χ_{1} ⊠ χ_{2} ⊠ χ of the Borel group B attached to three characters χ_{1}, χ_{2}, χ, defined by

( \begin{matrix}x & 0 & ∗ & ∗ \\ ∗ & y & ∗ & ∗ \\ 0 & 0 & t/x & ∗ \\ 0 & 0 & 0 & t/y \end{matrix} ) ↦ χ_{2}(x)χ_{1}(y)χ(t) .

We denote this induced representation by

χ_{1} × χ_{2} ◁ χ ,

where we use the (short) notation of [T]. For the reader who wants to compare with the results in [T] it should be remarked, that Tadic uses a slightly different realisation of the symplectic group of similitudes, which differs by a conjugation in Gl(4, F). If we take care of this and use corresponding choices of Borel groups, then the definition above matches the definition in [T].

We have the obvious isomorphism

χ_{1} × χ_{2} ◁ χ ≅ σ_{1} ◁ χ

where

σ_{1} = χ_{1} × χ_{2} := Ind_{B_{Gl(2)}^{Gl(2)}}(χ_{1}, χ_{2}) ≅ Ind_{\overline{B}}{Gl(2) \atop Gl(2)}(χ_{2}, χ_{1}) ≅ \frac{χ_{1}}{χ_{2}} ◁ χ_{2} ,

and where B_{Gl(2)} is the group of upper triangular matrices in Gl(2, F). This is consistant with the short notation. Also observe χ_{0}◁χ ≅ χχ_{0}×χ ≅ χ×χ_{0}χ ≅ χ_{0}◁χ_{0}χ for χ_{0}^{2} = 1, χ_{0} ≠ 1.

Furthermore we have the obvious isomorphism

χ_{1} × χ_{2} ◁ χ ≅ Ind_{Q}^{G}(χ_{1} ⊠ ρ) ,

where

ρ = χ_{2} ◁ χ = Ind_{B_{Gl(2)}^{Gl(2)}}(χ_{2}χ, χ) .

This is again consistant with the short notation.

The nonunitary principal series representations χ_{1} × χ_{2} ◁ χ of the group GSp(4, Q_{v}) are irreducible unless χ_{1}χ_{2} = ν^{±1} or χ_{1}/χ_{2} = ν^{±1} or χ_{1} = ν^{±1} or χ_{2} = ν^{±1} holds, where ν = |.|. See [T], thm 7.9 and remark 7.10. Observe

(χ_{1} × χ_{2} ◁ χ)⊗χ′ = χ_{1} × χ_{2} ◁ χχ′

ω_{χ_{1}×χ_{2}◁χ} = χ_{1}χ_{2}χ^{2} .

Furthermore one has the following equalities

χ_{1} × χ_{2} ◁ χ = χ_{2} × χ_{1} ◁ χ
χ_{1} × χ_{2} ◁ χ = χ_{1} × χ_{2}^{−1} ◁ χ_{2}χ

valid in the Grothendieck group R_{ZZ} (G) of representations of G = GSp(4, Q_{v}). They allow to reduce the determination of possible composition factors to the study of the cases

χ_{1} = ν or χ_{1}/χ_{2} = ν.

According to a result of Rodier (see [T], remark 8.3) the representation χ_{1} × χ_{2} ◁ χ is multiplicity free, if the characters (χ_{1}, χ_{2}, χ) involved are \underline{regular}. Regularity means in our case χ_{1} ≠ χ_{2}^{±1} and χ_{1} ≠ 1 and χ_{2} ≠ 1. Namely condition (iii) of Tadic 8.1 (b) is void in case n=2. In the regular case there are either 1 or 2 or 4 constituents.

No relatives are contained in unitary principal series, as they are irreducible (e.g. [T], corr. 7.6).

Discrete series occur in (nonunitary) principal series representations with regular characters only in the cases ν^{2} × ν ◁ χ or νχ_{0} × χ_{0} ◁ χ for χ_{0}^{2} = 1, χ_{0} ≠ 1. This is understood up to the obvious modifications under the action of the Weyl group described above. There is always a unique discrete series subquotient in these principal series representation denoted by δ(ν^{2} × ν ◁ χ) respectively δ(νχ_{0} × χ_{0} ◁ χ). See [T], thm 8.5.

The Borel group B (Nonregular principal series)

In this section the remaining nonregular cases of nonunitary principal series representations are discussed. See also [Rod], [ST] and [T]. Up to the allowable rearrangements of characters in the Grothendieck group, there are only the following three cases

a) ν^{1/2}χ_{0} × ν^{−1/2}χ_{0} ◁ χ , χ_{0}^{2} = 1.
b) 1 × ν ◁ χ
c) ν × ν ◁ χ

which remain to be considered.

For the discussion of these three cases we use the ”shuffle” formulas ([BDK], lemma 5.4). They allow to compute the Jaquet modules of induced representations. In the following \underline{exponents} of representations V are understood to be the constituents of the normalized Jaquet module r_{BG}(V) = δ_{B}^{−1/2} ⊗ V (B). These exponents are characters of the the diagonal split torus of G. Their absolute values will be listed in root-coordinates as point in the euklidean space R^{2} spanned by the roots and sometimes also called real exponents.

Let α_{1}, α_{2} be the short resp. long root of Sp(4). Let w_{max} the element of maximal length in the Weyl group and let s_{α_{1}}, s_{α_{2}} denotes the simple reflections attached to α_{1} and α_{2}. Define maps from R_{ZZ} (B) to R_{ZZ} (B) by

s_{α_{1}}(χ_{1} ⊠ χ_{2} ⊠ χ) = χ_{2} ⊠ χ_{1} ⊠ χ
s_{α_{2}}(χ_{1} ⊠ χ_{2} ⊠ χ) = χ_{1} ⊠ χ_{2}^{−1} ⊠ χ_{2}χ .

Then in our case the shuffle formulas are

r_{P G} ∘ i_{GP} (V) = V + w_{max}s_{α_{1}}(V) + i_{P B} ∘ s_{α_{2}} ∘ r_{BP} (V) .

Note w_{max}s_{α_{1}}(σ_{1} ⊠ χ) = σ_{1}^{∗} ⊠ ω_{σ_{1}}χ.

r_{QG} ∘ i_{GP} (V) = i_{QB} ∘ r_{BP} (V) + i_{QB} ∘ s_{α_{1}}s_{α_{2}} ∘ r_{BP} (V)
r_{QG} ∘ i_{GQ}(V) = V + w_{max}s_{α_{2}}(V) + i_{QB} ∘ s_{α_{1}} ∘ r_{BQ}(V) .

Note w_{max}s_{α_{2}}(χ ⊠ ρ) = χ^{−1} ⊠ ρχ.

r_{P G} ∘ i_{GQ}(V) = i_{P B} ∘ r_{BQ}(V) + i_{P B} ∘ s_{α_{2}}s_{α_{1}} ∘ r_{BQ}(V) .

For the conveniance of the reader let me consider the special and trivial representations Sp and 1_{Gl(2)} of Gl(2). Let denote Sp the discrete series representation constituent of

ν^{1/2} ×ν^{−1/2} = ν ◁ν^{−1/2} and Sp(χ) ≅ Sp⊗(χ∘det) the one of χν^{1/2} ×χν^{−1/2}. The normalized exponents are r_{Gl(2)B_{Gl(2)}}(Sp) = δ_{B_{Gl(2)}}^{1/2} = ν^{1/2} ⊠ ν^{−1/2} and r_{Gl(2)B_{Gl(2)}}(1_{Gl(2)}) = δ_{B_{Gl(2)}}^{−1/2}.

\underline{Nonregular case a)} (Discussion of π = ν^{1/2}χ_{0} × ν^{−1/2}χ_{0} ◁ χ with χ_{0}^{2} = 1):

Representations of type a) have two irreducible constituents Π and J. One is Π = Sp(χ_{0}) ◁ χ. It is tempered (up to the χ-twist) but not elliptic, whereas J is not tempered.

\underline{Proof}: By a twist we may assume χ to be unitary. Partial induction with respect to P yields two composition factors of π, the tempered Π = Sp(χ_{0}) ◁ χ and J = χ_{0} ∘ det ◁ χ.

By the shuffle formula the P-Jaquet module of Π has the following three constituents:

r_{P G}(Π) = 2 · Sp(χ_{0}) ⊠ χ + (ν^{1/2}χ_{0} × ν^{1/2}χ_{0}) ⊠ χχ_{0}ν^{−1/2} ,

two special ones with real exponents \frac{1}{2}α_{1} and one with the multiplicity two real exponent \frac{1}{2}(α_{1} + α_{2}). The tempered unitary representation Π is completely reducible and cannot contain a discrete series representation δ (see [Ka], §1, lemma 4). Otherwise δ can be embedded into two different representations induced from unitary discrete series, one times induced from G, δ and one time induced from P, Sp(χ_{0}). Thich is impossible. Therefore every constituent has to contain one of the two ”nondiscrete” tempered real exponents \frac{1}{2}α_{1}. Thus there are at most two constituents, one of them must contain both real exponents \frac{1}{2}(α_{1} + α_{2}). Using the shuffle formula with respect to the other maximal parabolic group Q, we conclude that

r_{QG}(Π) = ν^{1/2}χ_{0} ⊠ (ν^{−1/2}χ_{0} ◁ χ) + ν^{1/2}χ_{0} ⊠ (ν^{1/2}χ_{0} ◁ χχ_{0}ν^{−1/2})

has two irreducible constituents. Each of them has real exponents \frac{1}{2}α_{1}, \frac{1}{2}(α_{1} + α_{2}) with multiplicities 1. Therefore any constituent having the real exponent \frac{1}{2}(α_{1} + α_{2}) already contains all the real exponents. Hence Π is irreducible with real exponents \frac{1}{2}α_{1}, \frac{1}{2}(α_{1} +α_{2}) each with multiplicity two.

Now consider J = χ_{0} ∘ det ◁ χ. Its restriction to the symplectic group Sp(4) gives the representation I(s, χ_{0}) of Kudla und Rallis [KR] in the special case s = 0. According to [KR], prop. 5.1 this is an irreducible representation of Sp(4). Hence also J is irreducible. According to [KR] , p.235 its real exponents are −1/2α_{1} = (−1/2, 1/2) and −1/2(α_{1} +α_{2}) = (−1/2, −1/2) both with multiplicity two, where α_{1}, α_{2} are the short and long simple roots of B (in checking this one should be aware, that [KR] use a different choice of the minimal parabolic).

Observe χ_{0}ν^{1/2} × χ_{0}ν^{−1/2} ◁ χ has the same constituents as χ_{0}ν^{1/2} × χ_{0}ν^{1/2} ◁ χχ_{0}ν^{−1/2}. Thus it easily turns out, that J is up to a character twist by χχ_{0}ν^{−1/2} the Langlands quotient J_{P} (ν^{1/2}, (χ_{0} × χ_{0}) ⊠ 1) attached to a tempered representation of the Levi component of P.

\underline{Nonregular case b)} (Discussion of π = 1 × ν ◁ χ):

This case is the most interesting exceptional case. By a character twist we can (without restriction of generality) assume χν \frac{1}{2} to be unitary. One could even make it to one and then invoke Rodier’s result [Rod], §6, section 6.2. We will not do this. We will give an alternative proof.

In this case there will be four irreducible non isomorphic constituents, namely two irreducible nontempered representations J and J′ and two tempered nondiscrete representations T_{+} and T_{−}. Furthermore T_{+} and T_{−} are relatives in the tempered induced representation

T = T_{+} ⊕ T_{−} = 1 × Sp(χν^{\frac{1}{2}} ) .

The two representations T_{+}, T_{−} will also be denoted W p_{+}(ρ), W p_{−}(ρ) for ρ = Sp(χν \frac{1}{2} ). Remember Sp(χν \frac{1}{2} ) is the special representation twisted by χν \frac{1}{2} .

\underline{Definition}: This notion thus extends the definition of W p_{±}(ρ) from the cuspidal case to the case of all discrete series representations of Gl(2, Q_{v}).

\underline{Proof}: Observe

0 → Sp → ν^{1/2} × ν^{−1/2} → 1 ∘ det → 0 .

Partial induction from Q gives two constituents, the tempered constituent T = 1×Sp(χν \frac{1}{2} ) and the constituent 1 × (χν \frac{1}{2} ∘ det) with

0 → 1 × Sp(χν^{\frac{1}{2}} ) → 1 × ν ◁ χ → 1 × (χν^{\frac{1}{2}} ∘ det) → 0 .

From the shuffle formula the real exponents of the tempered constituent T are computed as \frac{1}{2}α_{2} and α_{1} + \frac{1}{2}α_{2}, each appearing with multiplicity two. The real exponents of 1 × (χν \frac{1}{2} ∘ det) are the opposite ones. More precisely one has

r_{QG}(T) = 1 ⊠ Sp(χν^{1/2}) + 1 ⊠ Sp(χν^{1/2}) + ν ⊠ (1 ◁ χ)
r_{P G}(T) = 2 · (Sp(ν^{1/2} ⊠ χ)) + 2 · (ν^{1/2} ∘ det ⊠ χ)

with exponents 2 · (1 ⊠ ν ⊠ χ) + 2 · (ν ⊠ 1 ⊠ χ).

We conclude: 1)Every irreducible constituent of 1×ν ◁χ, which has one of the exponents \frac{1}{2}α_{2} or α_{1} + \frac{1}{2}α_{2}, is contained in T. 2) Secondly T cannot contain a discrete series representation, as T is induced from a unitary discrete series representation of Levi(Q). This follows from [Ka], §1, lemma 4b). Therefore every constituent of T must contain an exponent of type \frac{1}{2}α_{1}.

The representation π = 1 × ν ◁ χ has the same constituents as ν × 1 ◁ χ. The second representation gives rise to an exact sequence

0 → L → ν × 1 ◁ χ → R → 0

with the modules R = (ν \frac{1}{2} ∘ det) ◁ χ and L = Sp(ν \frac{1}{2} ) ◁ χ. R has a unique irreducible (nontempered) quotient, which is the unique irreducible Langlands quotient J = J_{Q}(ν, 1⊠ 1 ◁ χν \frac{1}{2} ) ⊗ ν^{− \frac{1}{2}} of ν × 1 ◁ χ. Let T_{−} denote the kernel of the projection from R to J

0 → T_{−} → R → J → 0 .

The module L = Sp(ν \frac{1}{2} ) ◁ χ also has a unique irreducible (nontempered) quotient, its Langlands quotient J′ = J_{P} (ν \frac{1}{2} , Sp ⊠ χ). Let T_{+} denote the kernel of the projection from L to J′

0 → T_{+} → L → J′ → 0 .

Remark: We also use the notations T_{+} = Θ_{+}(Sp), T_{−} = Θ_{−}(1), J = Θ_{+}(1) later.

\underline{Exponents of L}: The shuffle formula gives three irreducible constituents for r_{QG}(L)

r_{QG}(L) = ν ⊠ (1 ◁ χ) + 1 ⊠ Sp(ν^{1/2}χ) + 1 ⊠ (ν^{1/2}χ ∘ det)

with the exponents

(ν ⊠ 1 ⊠ χ + ν ⊠ 1 ⊠ χ) + 1 ⊠ ν ⊠ χ + 1 ⊠ ν^{−1} ⊠ νχ .

The real exponents of the three pieces are α_{1} + \frac{1}{2}α_{2}, resp. \frac{1}{2}α_{2} resp. −\frac{1}{2}α_{2}. The last two appear with multiplicity one, the first one with multiplicity two. We conclude: The constituent T′ of L with exponent α_{1} + \frac{1}{2}α_{2} must contain it with multiplicity two, because they belong to the irreducible representation ν ⊠ (1 ◁ χ) of Levi(Q). By comparison with the discussion of T, we see that T′ is contained in T, therefore cannot have the exponent −\frac{1}{2}α_{2}. It cannot be a discrete series constituent, therefore the tempered constituent T′ contains also the exponent \frac{1}{2}α_{2}. It follows, that L has two irreducible constituents. One is T_{+} with real exponents 2(α_{1} + \frac{1}{2}α_{2}) and \frac{1}{2}α_{2}. which are different from The other is the Langlands quotient J′ with the real exponent −\frac{1}{2}α_{2}.

Finally

r_{P G}(L) = 2 · Sp(ν^{1/2}) ⊠ χ + Sp(ν^{−1/2}) ⊠ νχ + (ν^{1/2} ∘ det) ⊠ χ

has three constituents with exponents

ν ⊠ 1 ⊠ χ + 1 ⊠ ν^{−1} ⊠ νχ + (ν ⊠ 1 ⊠ χ + 1 ⊠ ν ⊠ χ) .

The corresponding real exponents are α_{1} + \frac{1}{2}α_{2} and −\frac{1}{2}α_{2} and \frac{1}{2}α_{2}, α_{1} + \frac{1}{2}α_{2}.

Note, that our preceeding discussion implied, that R and therefore T_{−} must contain an irreducible tempered constituent T/T_{+} with real exponent \frac{1}{2}α_{2}. Also J must contain the two exponents −α_{1} − \frac{1}{2}α_{2}. But instead of giving a similar discussion for the module R = (ν \frac{1}{2} ∘ det) ◁ χ as above, we invoke some

\underline{Results of Kudla and Rallis [KR]}:

The comparison with their results seems to be of independent interest.

We now work temporarily with the group Sp(4, F) instead of GSp(4, F). This is of no relevance, because the exponents in the case b) can be already distinguished on that level.

The Sp(4, F) analogue of the module R can be identified with the module

R = R_{2}(2, 2) = R_{2}(V_{1})

in the notation of [R-K]. V_{1} is the space underlying a split rank 4 quadratic form. By [KR], page 211-212, secondond theorem, part (v) the module R is isomorphic to the induced representation I_{2}(1/2, 1). The intertwining operator M_{2}^{∗}(1/2, 1) : I_{2}(1/2, 1) → I_{2}(−1/2, 1) (loc. cit. p.212) maps the maximal quotient J of R to the irreducible maximal submodule R_{2}(1, 1) of I_{2}(−1/2, 1). See loc. cit. p.211 , thm. 2 (i). It therefore has a composition series of lenght two with a unique irreducible quotient R_{2}(1, 1), therefore

J = R_{2}(1, 1)

and a unique irreducible subrepresentation R_{2}(4), therefore

T_{−} = R_{2}(4) = R_{2}(V_{2}) .

V_{2} is the quternionic quadratic space of rank 4.

For the conveniance of the reader we give some more details. One has to consider the theorem in loc.cit for the trivial character χ_{V} and n = 2, s_{0} = ±1/2 in the cases

(i) m = 2, O(1, 1)

and

(v) m = 4 , where V_{1} = V (2, 2) is split and V_{2} = V (4) is quaternionic anisotropic,

both having trivial character χ_{V}. Finally choose V_{1,0} = V (1, 1) to be split two dimensional, also with trivial character χ_{V}. Observe that R_{2}(4) is irreducible ([KR] prop 6.10), contained in R_{2}(2, 2) = I(1/2, 1) = Ind_{P}^{G}(ν^{1/2} ∘ det ⊠ χ) = R ([KR] prop. 3.4. (iii) and also p.254 bottom). The quotient is irreducible and isomorphic to R_{2}(1, 1). See p. 255 and p. 211 (i).

From loc. cit. prop. 4.5 and 4.6 we get the real exponents of the two constituents:

(0, 1) for R_{2}(4) and (−1, 0), (−1, 0), (0, −1) for J = R_{2}(1, 1). This is the notation of [KR] and

corresponds to the real exponents after the identifications α_{1} = (1, −1) and α_{2} = (0, 2).

Hence the real exponents are \frac{1}{2}α_{2} and −α_{1}−1/2α_{2}, −α_{1}−1/2α_{2}, −1/2α_{2} respectively. Thus T_{−} is tempered. This completes our sketchy argument concerning the group Sp(4, F).

Let us come back to our starting point:

Let now be the group be GSp(4, F) again. We deduced by a comparison of exponents that T = T_{−} has to be a constituent of T. Hence

T = T_{−} ⊕ T_{+} .

We furthermore found

r_{QG}(T_{+}) = ν ⊠ (1 ◁ χ) + 1 ⊠ Sp(χν^{1/2})

and

r_{QG}(T_{−}) = 1 ⊠ Sp(χν^{1/2}) .

It follows, that

r_{QG}(T_{+} − T_{−}) = ν ⊠ (1 ◁ χ)

with exponents 2 · (ν ⊠ 1 ⊠ χ). By inspection of r_{P G}(T) this implies

r_{P G}(T_{+} − T_{−}) = 2(Sp(ν^{1/2}) ⊠ χ) .

\underline{Remark}: We will later on need to know the exponents of the \underline{discrete} series representations found by Tadic, which are contained in the regular principal series representations. They are contained in χ_{0} × χ_{0}ν ◁ χ. Therefore it seems usefull, to note that an analogous discussion as above can be given for the representations χ_{0} × χ_{0}ν ◁ χ, with χ_{0}^{2} = 1, χ_{0} ≠ 1 (regular case). Especially we can also define the analog of the modules L, T_{+} and J, as introduced above. The discrete series representations δ(νχ_{0} × χ_{0} ◁ χν^{−1/2}) are contained in the correspondly defined module T_{+}, with precisely one real exponent α_{1} + \frac{1}{2}α_{2} (with multiplicity two). It is then clear from the discussion above, that

r_{QG}(δ(νχ_{0} × χ_{0} ◁ χν^{−1/2})) = νχ_{0} ⊠ (χ_{0} ◁ χν^{−1/2})
r_{P G}(δ(νχ_{0} × χ_{0} ◁ χν^{−1/2})) = Sp(ν^{1/2}χ_{0}) ⊠ χν^{−1/2}) + ·Sp(ν^{1/2}χ_{0}) ⊠ χχ_{0}ν^{−1/2})
r_{BG}(δ(νχ_{0} × χ_{0} ◁ χν^{−1/2})) = νχ_{0} ⊠ χ_{0} ⊠ χ_{0}χν^{−1/2}) + νχ_{0} ⊠ χ_{0} ⊠ χν^{−1/2}) .

This is shown using similar arguments as in the discussion of L above. Details will be left to the reader.

\underline{Nonregular case c)} (Discussion of π = ν × ν ◁ χ):

In this case there are two irreducible nonisomorphic nontempered constituents J, J′. Furthermore our discussion will imply AD_{χ} = 0 for χ = (B, ν × ν ◁ χ).

That there are exactly two irreducible constituents follows from [Rod], 6.3 in the case χ = 1, and then by character twists in the general case.

It remains to show the remaining statements. The representation ν × ν ◁ χ has a unique irreducible Langlands quotient J = J_{P} (ν, (1 × 1) ⊠ χ). Thus we have

ν × ν ◁ χ → J → 0

and J is nontempered.

One has an exact sequence

0 → L → ν × ν ◁ χ → R → 0 ,

where the module L = ν × Sp(χν \frac{1}{2} ) has a unique irreducible Langlands quotient J′ =

J_{Q}(ν, 1 ⊠ Sp) ⊗ χν^{\frac{1}{2}}

L → J′ → 0 ,

which is nontempered and not isomorphic to J.

The module R on the right side admits a nontrivial map

R → J → 0 .

So there are at least two irreducible nontempered constituents and we are done.

7.Resume:

We determined the Langlands quotients in all situations, where the induced representation is reducible and ”ramified” (with respect to the Weyl group).

Furthermore we found all elliptic tempered representations, which do not belong to the discrete series. A tempered representation is called elliptic, if its character is nontrivial on the regular elliptic locus of the group G. These are the limits of discrete series given as follows:

\underline{Lemma 6:} The limits of discrete series (tempered elliptic but not discrete series) are the representations W p_{+}(ρ), W p_{−}(ρ), where ρ is an irreducible discrete series representation

of Gl(2, Q_{v}).

Remember: W p_{±}(ρ) are the two irreducible constituents of the tempered representation 1 × ρ for irreducible discrete series representations ρ.

\underline{Proof}: The unitary principal series are irreducible, hence not elliptic. Therefore it is enough to consider representations of G, which are induced from discrete series representations of one of the two maximal parabolic subgroups. In the cases where this induction is from cuspidal representations, the only possibly candidates for elliptic families are the representations W p_{+}(ρ), W p_{−}(ρ) found by Waldspurger. It remains to see, that they actually are elliptic. Here we, according to [Ka], prop.1 it is enough to see, that W_{±}(ρ) can not be linearly expressed in the Grothendieck group by properly induced representations. See also [Cl], p.116. The only induced representations containing W_{±}(ρ) are of the form W_{+}(ρ) + W_{−}(ρ) (in the Grothendieck group). Hence W_{±}(ρ) are elliptic form cuspidal ρ.

To complete our discussion it remains to consider the cases, where induction is from a special representation of the Levi component of a maximal parabolic subgroup. It is enough to consider the cases, where the induced representation is not irreducible. This leaves us to consider the principal series representations

χ_{u} × ν ◁ χ

(in the case of the parabolic Q) respectively

χ_{u}ν^{1/2} × χ_{u}ν^{−1/2} ◁ χ

(in the case of the parabolic P) for suitable choices of unitary characters χ_{u}. So let us look at the composition factors of these representations.

In the notation of Tadic we get in the regular case χ_{u} ≠ 1 (first case for Q) and χ_{u}^{2} ≠ 1 (second case for P), that his parameter s(ϕ) = 1 in both cases. Namely in the notation

of [T] we have s = 2, p = 0, t = 0, ϵ = 1 in the first case and s = 2, p = 1, t = 0, ϵ = 0 in

the second. Therefore in both these cases the underlying principal series representations induced from B to G have two irreducible constituents with multiplicity one. They are

χ_{u} ×Sp(χν^{1/2}) and χ_{u} ×(χν^{1/2} ∘det) in the first case respectively Sp(χ_{u})◁χ and (χ_{u} ∘det)◁χ

in the second. Therefore the corresponding induction of the special representations from the maximal parabolic groups gives irreducible representations. Hence the regular cases do not give limits of discrete series, because both irreducible constituents are itself induced from a proper parabolic subgroup. Thus their characters must vanish on the elliptic locus.

It remains to examine the nonregular choices χ_{u} = 1 in the first case for Q respectively χ_{u}^{2} = 1 in the second case for P. The decomposition series of the principal series representations for these cases was discussed in b) and a). In the case a), corresponding to the parabolic P, we had found only one irreducible tempered constituent Π, which itself was induced Π = Sp(χ_{0}) ◁ χ. Therefore this tempered representation has trivial character on the elliptic locus and is not a limit of a discrete series. In the case b) however, which corresponds to induction from special representations from the Levi component of the group Q, we found that the tempered representation 1 × Sp(χν \frac{1}{2} ) splits into two irreducible tempered constituents T_{+} and T_{−}. They are relatives in the sense of Kazhdan and possibly elliptic. It only remains to show, that they are elliptic. Again it is enough to show according to [Ka], prop. 1 that T_{±} can not be expressed as a linear combination of induced representations in the Grothendieck group. This follows from our discussion of the nonregular case 1b) in section 6. What can be combined by induced representations

must have the form α(T_{−} + J) + β(T_{+} + J′) + γ(T_{−} + T_{+}) + δ(J + J′) for α, β, γ, δ ∈ ZZ. This

shows, that T_{+} and T_{−} are tempered elliptic. This completes the proof of lemma 6.

A similar discussion determines all cases where AD_{χ} = 0 fails to hold. For an infinitesimal character χ = (L, ρ) one has AD_{χ} = 0 unless ρ is ”ramified” with respect to the Weyl group. So only irregular principal series are relevant and furthermore also the representations induced from cuspidal representations of Levi components of the two maximal parabolic subgroups. Our discussions of exponents in the cases a), b) and c) and the results of Shahidi and Waldspurger show

\underline{Lemma 7:} Asymptotic independency only fails for pairs χ = (L_{Q}, 1⊠ρ), where ρ is a cuspidal representation of Gl(2, F). There χ corresponds to the two relatives W p_{+}(ρ), W p_{−}(ρ)

with AD_{χ} = ZZ(W p_{+}(ρ) − W p_{−}(ρ)).

\underline{Proof}: The case where ρ is special does not occur. In that case T_{+} = W p_{+}(ρ) and T_{−} = W p_{−}(ρ) have different exponents as shown in the discussion of case b).

\underline{List of unitary discrete series representations of G}:

The unitary discrete series representations of GSp(4, Q_{v}) are given by the following representations

1) The unitary cuspidal representations of G

The second types arise as constituents of representations induced from the maximal parabolic P (Siegel parabolic)

2) The series δ(σ_{1}ν^{1/2} ◁ χν^{−1/2}) for σ_{1} a discrete series representation for Gl(2) with

ω_{σ_{1}} = 1 and χ unitary, but σ_{1} ≠ Sp.

Observe that except the cases, where σ_{1} is cuspidal, the cases, where σ_{1} = Sp(χ_{0}) arise from discrete series coming from regular principal series representations, were found by

Tadic. They are the δ(Sp(χ_{0})ν^{1/2} ◁ χν^{−1/2}) = δ(νχ_{0} × χ_{0} ◁ χν^{−1/2}) for χ_{0} ≠ 1, χ_{0}^{2} = 1 and

unitary χ. Note the ”missing” discrete series corresponding to the forbidden choice σ_{1} = Sp. In this case one only has the ”virtuel” discrete series counterpart T_{+} − T_{−} defined in the Grothendieck group R_{ZZ} (G).

The third type of discrete series representations arises from constituents of representations induced from the maximal parabolic Q

3) The series δ(νχ_{0} × σν^{−1/2}) for unitary (cuspidal) discrete series representations σ of Gl(2) with CM by χ_{0}, i.e. χ_{0} ≠ 1, χ_{0}^{2} = 1 and σ ⊗ χ_{0} ≅ σ.

Finally there are

4)l The Steinberg representations δ(ν^{2} × ν ◁ χν^{−3/2}) for unitary χ.

8. Asymptotics

\underline{Parabolic subgroups of M}:

Consider the group

M = Gl(2, F) × Gl(2, F)/{(t, t^{−1}) | t ∈ F^{∗}} .

We fix parabolic subgroups in M

B_{M} = \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) × ( \begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix} ) mod F^{∗}\}

respectively

P_{M} = \{( \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} ) × ( \begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix} ) mod F^{∗}\} .

They are representatives of the proper parabolic subgroups of M modulo the action of the group of the automorphism group of M. The simple roots corresponding to this choice of the Borel subgroup are given by the characters, which map diag(t_{1}, t′_{1}), diag(t_{2}, t′_{2})) mod F^{∗} to t_{1}/t′_{1} respectively t′_{2}/t_{2}.

\underline{Parabolic subgroups of G = GSp(4, F)}:

We fix the Borel group B ⊂ G

B = \{( \begin{matrix}∗ & 0 & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \\ 0 & 0 & 0 & ∗ \end{matrix} ) ∈ G\}

and the Siegel parabolic P and the second maximal proper parabolic subgroup Q containing B. The simple positive roots correspond to the two characters, which map diag(x_{1}, x_{2}, x′_{1}, x′_{2}) ∈ B to x_{2}/x_{1} (short root) respectively x_{1}/x′_{1} (long root).

\underline{Levi components}:

We will identify the Levi components of the parabolic subgroups P_{M} ⊂ M and P ⊂ G via

the map ψ : Levi(P_{M}) → Levi(P)

ψ((x, diag(t_{2}, t′_{2})) mod F^{∗}) = ( \begin{matrix}t′_{2}det(x)x^{−t} & 0 \\ 0 & t_{2}x \end{matrix} ) .

This induces identifications Levi(B_{M}) → Levi(B)

(diag(t_{1}, t′_{1}), diag(t_{2}, t′_{2})) mod F^{∗} ↦ diag(t′_{2}t′_{1}, t′_{2}t_{1}, t_{2}t_{1}, t_{2}t′_{1})

and conversely

diag(x_{1}, x_{2}, x′_{1}, x′_{2}) ↦ (diag(x_{2}, x_{1}), diag(x′_{1}/x_{2}, 1)) mod F^{∗} .

The maximal tori of G, which come from M via an admissable embedding and in addition are contained in a proper parabolic subgroup of G, are of type 6) or 7) as given in our list of tori. Remember, these cases are

6) T ≅ L^{∗} × F^{∗} ⊂ P, where the further data are ϵ(T) = 2, W_{F} (T, G) = W_{F} (T, M) and

D_{G}(T) = 1.

7) T ≅ (F^{∗})^{3} ⊂ B, where the further data are ϵ(T) = 1, W_{F} (T, M) = W_{M} is a subgroup

of index two in W_{F} (T, M).

Among the maximal tori in Levi(Q) only the split tori come from M. They can be studied already in B.

\underline{Character Lift}:

We had the following formula

α^{G}(t^{δ}) = D_{G}(t)^{−1}ϵ(T) ∑_{w∈W_{F} (T,M)\W_{F} (T,G)} τ(wtw^{−1})κ(w(δ))D_{M}(wtw^{−1})α(wtw^{−1}) .

for semisimple regular t ∈ G.

Assume α to be a distribution on M, which is invariant under conjugation and under the outer automorphism ∗. Put f = α^{G}. We will assume in the following, that α is a finite linear combination of irreducible characters. Then f = α^{G} is a locally finite sum of characters as shown already. Therefore the Jaquet coefficients f(P) for parabolic subgroups P of G are defined.

The formula for the character lift for ∗-invariant α and the tori T contained in parabolic subgroups of G, which come from M (cases 6 and 7) simplifies to

D_{G}(t)α^{G}(t) = 2 · D_{M}(t)α(t) , t ∈ T .

Remember D_{G}(t) = ∏_{α} |α(t) − 1|^{1/2} for t ∈ T semisimple regular, where the product runs over all roots α of the maximal torus T.

For the discussion of character lifts χ_{σ_{v}}^{G} we already gave some discussion for cuspidal representations σ_{v}. Also we already discussed the case of induced representations. Hence it remains to consider the special representations, i.e the lift of representations of the form Sp(χ) ⊗ ρ, ω_{ρ} = χ^{2} from M to G. Every special representation Sp(χ) = Sp ⊗ χ is obtained from the Steinberg representation Sp by a twist. Because the character lift commutes with twists by one dimensional characters (lemma 1), we can assume χ = 1 and hence ω_{ρ} = 1. Thus we only need to consider the two cases

1) ρ_{v} ≅ (Sp ⊗ ρ) ⨁(ρ ⊗ Sp)

where ρ is a cuspidal irreducible representation of Gl(2, F) with trivial central character ω_{ρ} = ω_{Sp} = 1 or, alternatively, where ρ_{v} ≅ Sp(χ_{0}) and χ_{0}^{2} = 1, χ_{0} ≠ 1. Then there is the case

2) ρ_{v} ≅ Sp ⊗ Sp.

Observe, that we made α = χ_{ρ_{v}} invariant under ∗ in the first case.

In the \underline{first case} put

f = α^{G} − 2 · χ_{δ(ρν^{1/2}◁ν^{−1/2})} .

This is well defined, because ω_{ρ} = 1. In general one has to attach to the representation Sp(χ) ⊗ ρχ of the group M the representation δ(ρν \frac{1}{2} ◁ χν^{−1/2}).

In the \underline{second case} put

f = α^{G} − χ_{(T_{+}−T_{−})} ,

where T_{+} ⊕ T_{−} = 1 × Sp, the tempered constituents of 1 × ν ◁ ν^{−1/2}.

\underline{Lemma 8}: Then f(P) = f(B) = f(Q) = 0 (Jaquet modules).

\underline{Remark}: In the first case we could consider α = χ_{ρ⊗Sp} instead. The lemma then implies (α^{G} − χ_{δ(ρν^{1/2}◁ν^{−1/2})})(P) = 0 for proper parabolic subgroups P of G.

\underline{Proof}: We have to show r_{PG}(f) = 0 for proper parabolic subgroups P of G, where

r_{PG} : RZZ (G) → RZZ (Levi(P))

is the normalized Jaquet functor of [Ka]. It is enough to consider maximal parabolic subgroups P.

\underline{Claim}: For semisimple regular ag with ag ∈ P and a ∈ A_{P} sufficantly P-contractive and g in a fixed compact set Ω one has

r_{PG}(α^{G})(ag) = (D_{G}α^{G})(ag) .

It is enough to show

χ_{r_{PG}(π)}(ag) = (δ_{P}^{−1/2}χ_{π(P)})(ag) = (D_{G}χ_{π})(ag) = (δ_{P}^{−1/2}χ_{π})(ag)

Let g be semisimple regular with g ∈ T. Then there exists an integer n such that g^{n} = a·s, where a ∈ A_{θ} and s ∈ S, where S ⊂ T is anisotropic and T = A_{θ} · S. Remember Δ was a fixed basis of simple roots, P = P_{θ} for θ ⊂ Δ a standard parabolic with split component A_{θ}. Then for every root α of T we get |α(g)| = |a · s|^{1/n} = |a|^{1/n}. Now suppose a ∈ A_{P} to be P-contractive and g to vary in some fixed compact set Ω ⊂ P. Then we get for a sufficantly P-contractive

D_{G}(ag) = D_{Levi(P)}(g)δ_{P}^{−1/2}(a) .

This together with lemma 5 proves the claim.

The similar formula applies for the groups M under consideration. Therefore lemma 5 reduces the proof to the calculation of the normalized Jaquet functors. There is some small difficulty to keep track of asymptotic conditions, when elements ag ∈ T are considered as elements of the two different groups M and G. We will treat the two cases separately.

\underline{Discussion of the first case}(ρ cuspidal):

We begin with the \underline{Siegel parabolic}. We identified the Levi components of P_{M} and P via the isomorphism ψ. Using this we get δP_{M}^{\frac{1}{2}} (t) = |t′_{2}/t_{2}| = δ_{P} (t)^{1/6} for elements The Siegel parabolic is the standard parabolic group attached to the short root α_{1}. For

t = (x, diag(t_{2}, t′_{2})) mod F^{∗} in P_{M}. Thus δ(ρν^{1/2} ◁ ν^{−1/2}) = δ(ρδ_{P}^{1/6} ◁ ν^{−1/2}).

t = ga with g ∈ Ω ∩ P and a ∈ A_{P} and a = diag(t′_{2}c, t′_{2}c, t_{2}c, t_{2}c) with |α_{2}(a)| = |t′_{2}/t_{2}| <

ϵ(Ω) << 1 we therefore get from lemma 5

r_{P G}(χ_{δ(ρν^{1/2}◁ν^{−1/2})})(t) = χ_{ρν^{\frac{1}{2}} ⊠ν^{−1/2}}(ag) , |α_{2}(a)| << 1 .

This follows, because δ(ρν^{1/2} ◁ν^{−1/2}) is one of two irreducible constituents of the induced representation ρν^{1/2} ◁ν^{−1/2}. The normalized Jaquet module of the induced representation has two irreducible constituents ρν^{1/2} ⊠ ν^{−1/2} and ρν^{−1/2} ⊠ ν^{1/2}. The discrete series representation corresponds to the tempered exponent: r_{P G}(δ(ρν^{1/2}◁ν^{−1/2})) = ρν^{1/2}⊠ν^{−1/2}. Thus for sufficantly P-contractive elements a and g ∈ Ω and t = ag

r_{P G}(δ(ρν^{1/2} ◁ ν^{−1/2}))(t) = χ_{ρν^{1/2}}(t_{2}det(x)x^{−t})ν^{−1/2}(t_{2}t′_{2}det(x)) = χ_{ρ}(x)|t′_{2}/t_{2}|^{1/2} ,

where we used ω_{ρ} = 1.

On the other hand

r_{P G}(α^{G})(ag) = (D_{G}α_{G})(ag)
= 2 · D_{M}(ag)(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(ag) = 2 · r_{P_{M} M}(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(ag)

holds if |α_{2}(a)| = |t′_{2}/t_{2}| << 1. This is equivalent to the condition, that |β(ψ^{−1}(a))| = |t′_{2}/t_{2}| << 1, where β is the positive root of P_{M}. Therefore

r_{P G}(α^{G})(ag) = 2 · r_{P_{M} M}(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(ag)

= 2 · r_{P_{M} M}(χ_{ρ⊗Sp})(ag)
= 2 · χ_{ρ⊗δ_{P_{M}}^{1/2}} (ag) = 2χ_{ρ}(x)|t′_{2}/t_{2}|^{1/2}

if |β(a)| << 1 with g in a fixed compact subset Ω. Hence

(r_{GP} (f))(ag) = 0

holds for all such a. By lemma 5 this implies r_{GP} (f) = 0, equivalently f(P) = 0. Especially we get f(B) = 0.

Digression (ρ = Sp(χ_{0})): Let us briefly indicate, which modifications have to be made in the noncuspidal case ρ = Sp(χ_{0}). Put δ = δ(ρν^{1/2} ◁ ν^{−1/2}. Firstly it turns out that

r_{P G}(δ)(t) = (1 + χ_{0}(t_{2}t′_{2}det(x))χ_{ρ}(x)|t′_{2}/t_{2}|^{1/2} ,

because r_{P G}(δ) = ρν \frac{1}{2} ◁ ν^{−1/2} + ρν \frac{1}{2} ◁ χ_{0}ν^{−1/2}. Secondly we get the same conclusion as above because of

r_{P G}(α^{G})(ag) = 2 · r_{P_{M} M}(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(ag)
= 2χ_{ρ}(x)|t′_{2}/t_{2}|^{1/2} + 2χ_{Sp}(x)χ_{0}(t_{2}t′_{2})|t′_{2}/t_{2}|^{1/2} ,

which coincides with r_{P G}(δ)(ag).

The \underline{second maximal parabolic} Q: It remains to show f(Q) = 0 or equivalently r_{QG}(f) = 0. It is clear from the definition of α^{G} and δ(ρν^{1/2} ◁ ν^{−1/2}), that the difference f(t) vanishes for elliptic semisimple regular elements t in Q. It remains to consider the split regular elements t corresponding to matrices

t = (diag(t_{1}, t′_{1}), diag(t_{2}, t′_{2})) mod F^{∗} ∈ B_{M}

or its image (suppressing ψ)

t = diag(t′_{1}t′_{2}, t_{1}t′_{2}, t_{1}t_{2}, t′_{1}t_{2}) ∈ B .

We have

(r_{QG}f)(t) = (r_{QG}α^{G})(t) − 2 · χ_{r_{QG}(δ(ρν^{1/2}◁ν^{−1/2}))}(t)
= (r_{QG}(α^{G}))(t) .

Namely r_{QG}(δ(ρν^{1/2} ◁ν^{−1/2})) = 0 is a consequence of the shuffle formula. In order to show r_{GQ}(f)(t) = 0 it is furthermore enough to consider such elements t = ag with the property |α_{1}(a)| << 1 and g in some fixed compact subset of Q (lemma 5). This extra condition

implies |α_{1}(t)| = |t_{1}/t′_{1}| << 1 and 0 < const_{1} < |t′_{1}t′_{2}/t_{1}t_{2}| ≤ const_{2} for all such t. In other

words |t′_{2}/t_{2}| < const_{2}|t_{1}/t′_{1}| << 1. But then for such elements t = ag with the property |α_{1}(a)| << 1 and g in some fixed compact subset of Q.

(r_{QG}f)(t) = (r_{QG}α^{G})(t) = D_{G}(t)α_{G}(t)
= 2 · D_{M}(t)(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(t) .

As we have seen t = ag is then also B_{M} contractive. Therefore

(r_{QG}f)(t) = 2 · D_{M}(t)(χ_{ρ⊗Sp} + χ_{Sp⊗ρ})(t) = 0

for such t, because ρ is cuspidal. Equivalently this gives f(Q) = 0. This completes the proof of the lemma in the first case for cuspidal ρ.

In the non cuspidal case, if ρ = Sp(χ_{0}) and χ_{0}^{2} = 1, χ_{0} ≠ 1, then again it is enough to consider split regular elements t ∈ Q. A similar computation as above gives

r_{QG}(f)(t) = 2 · (χ_{r_{M_{P} M} (ρ⊗Sp+Sp⊗ρ)}) − 2 · χ_{r_{BG}(δ(ρν^{1/2}◁ν^{−1/2})} .

This is zero, because the term in both brackets become equal to χ_{0}ν ⊠ χ_{0} ⊠ ν^{−1/2}(t) +

χ_{0}ν ⊠ χ_{0} ⊠ χ_{0}ν^{−1/2}(t) or explicitely |t_{1}t′_{2}/t′_{1}t_{2}|^{1/2}(χ_{0}(t_{1}t′_{1}) + χ_{0}(t_{2}t′_{2}).

\underline{Discussion of the second case} (ρ = Sp):

Now consider the second case α = χ_{Sp⊗Sp}. From the shuffle formula and the knowledge of the exponents of T_{+} and T_{−} one gets

r_{P G}(T_{+} − T_{−}) = 2 · (Sp(ν^{\frac{1}{2}} ) ⊠ ν^{−1/2}) .

On the other hand we have D_{G}(t)α^{G}(t) = 2 · D_{M}(t)α(t) for all regular semisimple t ∈ G coming from M. Therefore, for (x, diag(t_{2}, t′_{2})) ∈ P_{M} with |t′_{2}/t_{2}| << 1 and x in a fixed compact set Ω, we have

r_{P G}(α^{G}) = 2 · r_{P_{M} M}(α) .

Therefore

r_{P G}(α^{G} − χ_{(T_{+}−T_{−})}) = 0

is a consequence of lemma 5 together with

r_{P_{M} M}(α)(x, diag(t_{2}, t′_{2})) = r_{P_{M} M}(χ_{Sp⊗Sp})(x, diag(t_{2}, t′_{2})) = χ_{Sp}(x)|t′_{2}/t_{2}|^{\frac{1}{2}}

and (as in the first case)

χ_{Sp(ν^{1/2})×ν^{−1/2}}(diag(t′_{2}det(x)x^{−t}, t_{2}x)) = χ_{Sp}(x)|t′_{2}/t_{2}|^{1/2} ,

which hold in the above given range of x, t_{2}, t′_{2}.

The remaining discussion of the vanishing f(Q) = 0 preceeds as in the first case. Observe, that there we only used two facts: Namely that f(B) = 0 holds, which follows from f(P_{Q}) = 0, and secondly that r_{QG}(f) is a character of Levi(Q), which vanishes on the elliptic regular elements. The analog of the first fact has already been established. The second statement is true for α^{G}. So it only remains to show it also for the character of r_{QG}(T_{+} − T_{−}). But T_{+} ⊕ T_{−} ≅ 1 × Sp, hence as we already know

r_{QG}(T_{+} − T_{−}) = ν ⊠ (1 ◁ ν^{−1/2}) .

Therefore r_{QG}(T_{+} − T_{−}) vanishes on the elliptic locus of Levi(Q) as we claimed, because it is given by the character of a representation induced from the Borel subgroup.

\underline{Corollary 7}: Suppose σ_{v} = σ_{1,v} ⊗ σ_{2,v} is an irreducible cuspidal series representation of M, which is not invariant under the outer automorphism ∗. Then either χ_{σ_{v}}^{G} is an integral locally finite linear combination

χ_{σ_{v}}^{G} = n(π_{v})χ_{π_{v}} + ∑_{ρ cuspidal} n(ρ)(χ_{W p_{+}(ρ)} − χ_{W p_{−}(ρ)}) ,

where π_{v} is irreducible cuspidal if σ_{v} is cuspidal and and n(π_{v}) ∈ {0, ±1} and where n(ρ) ∈ ZZ and ρ runs over the cuspidal representations of Gl(2, F); or there are two irreducible cuspidal representations π_{+}, π_{−} such that

χ_{σ_{v}}^{G} = χ_{π_{+}} − χ_{π_{−}} .

If σ_{v} is irreducible cuspidal and invariant under ∗, then

χ_{σ_{v}}^{G} = n(π_{v,1})χ_{π_{v,1}} + n(π_{v,2})χ_{π_{v,2}} + n(π_{v,3})χ_{π_{v,3}} + n(π_{v,4})χ_{π_{v,4}}

+ ∑_{ρ cuspidal} n(ρ)(χ_{W p_{+}(ρ)} − χ_{W p_{−}(ρ)}) ,

where the π_{v,i} are irreducible non isomorphic cuspidal representations with n(π_{v,i}) ∈ {1, 0, −1} and ρ are cuspidal representations of Gl(2, F) with n(ρ) ∈ ZZ.

\underline{Proof}: From lemma 3 we obtain χ_{σ_{v}}^{G} as an integral linear combination of characters of cuspidal discrete series representations and the W p_{+}(ρ) − W p_{−}(ρ) for cuspidal ρ’s. The remaining statement follows from the formula ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩_{G,e} = 2 resp = 4 by the method 1.

\underline{Corollary 8} : The similar statement follows if σ_{v} is discrete series but not cuspidal. In that case the statements carry over with the replacements n(π_{v}) = 1, π_{v} =

δ(ρν^{1/2}◁χν^{−1/2}) for σ_{v} = ρ⊗Sp(χ) and n(π_{v,1}) = 1, π_{v,1} = (T_{+}−T_{−})⊗χ for σ_{v} = Sp(χ)⊗Sp(χ).

\underline{9. Whittaker models}:

Let π = ⊗′π_{v} be a global cuspidal irreducible automorphic representation of GSp(4). Then π is called ψ-generic, if it has a global Whittaker model with respect to the character ψ. As any two nondegenerate characters ψ are conjugate in our case, we can omit to mention ψ. Especially all π_{v} are then locally generic, i.e have a Whittaker model. Conversely assume, that all local components π_{v} are generic. If the degree 5 standard L-function does not have a zero ζ(π, s) on the line Re(s) = 1, then π is globally generic ([AG], p.49).

Suppose, π is a weak lift from M attached to an irreducible cuspidal automorphic form σ with holomorphic infinite component σ_{∞} = σ_{1,∞} × σ_{2,∞} of weights k_{i} ≥ 2, then the σ_{i} are all tempered by the Ramanujan conjecture. Furthermore the degree 5 standard zeta function is ζ(π, s) = ζ(s)L(σ_{1} ⊗ σ_{2}^{∗}, s). It does not have a zero on the line Re(s) = 1. See [Sh2], [AC] p. 200. Therefore \underline{Theorem 1}: A weak lift from M is generic if and only if it is locally generic for all places v.

\underline{Theorem 2 ([S2], p.295)}: Two cuspidal, irreducible generic representations, which are locally isomorphic for almost all places v are isomorphic.

According to a theorem of Waldspurger, Moeglin and Rodier for an irreducible admissable representation σ_{v} of a quasisplit reductive group over a local field F_{v} the following are equivalent (see [Sh], p. 325)

(i) σ_{v} is generic

(ii) The germ expansion of the character of σ_{v} has a nontrivial (positive) contribution from at least one of the maximal (regular) nilpotent orbits.

This implies by the character formulas, that for generic irreducible representations on a (quasisplit) Levi component, the induced representation has at least one generic irreducible constituent. Note that the induced representation itself need not have a Whittaker model itself. Especially one of the representations W p_{±} has a Whittaker model. By a double coset consideration it is clear, that only one of them can be generic.

Another consequence is

\underline{Lemma 9}([Sh], prop. 9.6): For tempered irreducible generic representations σ_{v} of M_{v} at least one of the coefficients n(π_{v,i}) of r(σ_{v}) = ∑ n(π_{i})π_{v,i} (as in corollary 7 and 8) is positive n(π_{v,i}) > 0 with π_{v,i} generic for GSp(4, F_{v}).

\underline{Notation}: Fix one such generic π_{v,i} as in the lemma and let it denote

π_{+}(σ_{v}) σ_{v} generic, irreducible .

We will eventually see, that π_{+}(σ_{v}) is \underline{unique}.

In order to understand π_{+}(σ_{v}) we consider the local \underline{theta lift} from M = GSO(2, 2) to G = GSp(4), as in the work of Soudry [S3], [S4] p.363 ff and [PSS] (loc.cit p. 514 local theory). Note that GSO(2, 2) = (Gl(2) × Gl(2))/{(t, t^{−1}), t ∈ F^{∗}} is the connected component of the group of similitudes of a split quadratic form in four variables. The corresponding global lift was introduced in [S3] and [PSS2], p.416. If σ = σ_{1} × σ_{2} is a global \underline{generic} cuspidal automorphic representation of M, then this theta lift Π_{+}(σ) is an automorphic representation of GSp(4), which does not vanish and has a global Whittaker model. σ is geneneric, if it is non degenerate in the sense that dim(σ_{i}) ≠ 1 for i = 1, 2.

However (GSO(2, 2), GSp(4)) is not really a dual reductive pair in the sense of Howe. The corresponding dual reductive pair is (O(2, 2), Sp(4)). This was studied in [HPS]. The restriction of Π_{+}(σ) to Sp(4, A A) is a sum of representations of the type given by theorem 8.1 of [HPS] up to character twists μ in the notation of loc. cit.

The global representations Θ_{+}(σ) attached to generic cuspidal automorphic representations σ are cuspidal, if the corresponding theta lift from GSO(2, 2) to Gl(2) vanishes. This is shown as in [HPS], thm 8.1 and p.78 ff.. Assume, that the lift of σ to Gl(2) vanishes. This is the case if σ = σ_{1} × σ_{2} with σ_{1} non isomorphic to σ_{2}. Then each irreducible constituent π of this lift Π_{+}(σ) is a weak cuspidal lift (in our sense) of the automorphic cuspidal representation σ from M to GSp(4). This follows from [PSS], thm 2.4 or from formula (*) in [PSS2], p.417. Formula (*) implies that the degree 4 standard L series of π is given by

L_{v}(Θ_{+}(σ), s) = L_{v}(σ_{1}, s)L_{v}(σ_{2}, s)

at almost all places v. By [S] it follows, that such π is not CAP if σ is generic, i.e. if none of the σ_{i} is one dimensional. There is at least one irreducible cuspidal constituent π of Θ_{+}(σ), which has a global Whittaker model. Denote this constituent

Θ_{+}(σ) .

By the generic strong multiplicity one theorem above Θ_{+}(σ) is then uniquely determined by σ (generic with σ_{i} non isomorphic) and independent from any other choices made. We call it the \underline{global theta lift}. Its local components are

Θ_{+}(σ)_{v} .

They have local Whittaker models. These local representations are also uniquely determined by the local theta correspondence by their property to have a Whittaker model. See lemma 9 below.

One remark is order to understand this properly. According to the Howe conjecture, there should be a well defined local correspondence

σ_{v} ↦ Θ_{+}(σ_{v})

in our situation, where Θ_{+}(σ_{v}) is a unique irreducible quotient of the local theta lift Π_{+}(σ_{v}) studied in [S4]. The Howe conjecture is proved, if the characteristic of the residue field is different from 2. However we do not have a proper dual reductive pair and we also want to include the case odd residue characteristic 2. Therefore we remark, that there are two cases, where a local theta lift Θ_{+}(σ_{v}) can be uniquely defined.

\underline{Lemma 10}: Let σ_{v} be an irreducible admissable representation of M = GSO(2, 2)_{v}. Then up to isomorphism there is at most one irreducible quotient π_{v} (also called Θ_{+}(σ_{v})) of the local theta lift Π_{+}(σ_{v}) if the following is property of π_{v} is required. Either

1) σ_{v} = Ind(χ̃_{1}, χ̃_{2}) × Ind(χ_{1}, χ_{2}) is unramified and π_{v} is unramified. Then

π_{v} ≅ χ̃_{1}χ_{1}^{−1} × χ̃_{2}χ_{1}^{−1} ◁ χ_{1} .

2) or σ_{v} is generic and the quotient π_{v} is assumed to be generic.

\underline{Proof}: This follows from [PSS2], p.417 ff and from [S4], thm. 3.1.

This allows us to define a \underline{local theta lift}

σ_{v} irreducible, generic ↦ Θ_{+}(σ_{v}) irreducible, generic

for all generic σ_{v}, for which the theta representation Π_{+}(σ_{v}) has an admissable irreducible generic quotient. This local assignment commutes with character twists.

The theta representation Π_{+}(σ_{v}) has an admissable irreducible generic quotient, if σ_{v} is embeddable into a global generic cuspidal automorphic representation σ of M. This follows from the global theory. By a character twists this allows to define local theta lifts Θ_{+}(σ_{v}) for all discrete series representations σ_{v} of M_{v}. The local theta lift Θ(σ_{v}) is also defined for all representations

σ_{v} = Ind(χ̃_{1}, χ̃_{2}) × Ind(χ_{1}, χ_{2})

for \underline{unitary} characters χ̃_{i}, χ_{i}. For the spherical case see [PSS2], page 417. The argument given there extends to the case, where one of the characters is ramified. In this case the map ν constructed in [PSS], p.418 properly exists (not as a value of a meromorphic function). By Tate theory the integration over b and t in loc. cit. is welldefined in the sense of analytic continuation. We only have to assume χ̃_{2}/χ_{1}(a) ≠ 1, to make the integral of loc. cit well defined. Without restriction of generality this is the case unless χ̃_{1} = χ̃_{2} = χ_{1} = χ_{2}. Then this is a twist of the spherical case already considered. The map ν gives rise to a nontrivial intertwining operator

T_{v} : (Big Weil Representation) ⊗ σ_{v} → χ̃_{1}χ_{1}^{−1} × χ̃_{2}χ_{1}^{−1} ◁ χ_{1}

as in [PSS2], p.417. If the image of T_{v} has a generic quotient, then we are done. But by [T], corr. 7.6 the unitary principal series of GSp(4, F_{v}) are irreducible. So this finishes the argument.

There is one particular further case, which is of importance later

\underline{Lemma 11}: Let σ_{v} = σ_{2,v} ×σ_{2,v} with a cuspidal irreducible representation σ_{2,v} of Gl(2, F_{v}). Then

Θ_{+}(σ_{2,v} × σ_{2,v}) = W p_{+}(σ_{2,v}) .

\underline{Proof}: See [Wa], lemma 8.2, p.64 and the definition of Y′ in loc. cit. p.62-63. Then Y′ = D_{v}^{2} where D_{v} = M_{2,2}(F_{v}). The space D_{v} can be identified with the quadratic space of a split quadratic form, realized by the quadratic form det (the determinant) on the space D_{v} of 2 by 2 matrices. The group M = GSO(2, 2) operates by (h_{1}, h_{2}) ∘ x = h_{1}xh′_{2}

for (h_{1}, h_{2}) ∈ M and x ∈ D_{v}.

It is shown in [Wa], that the Weil representation admits a nontrivial map

ζ : S(D_{v}^{2} × F_{v}^{∗}) → Ind_{Q}^{G}(1 ⊠ ρ_{v}) ⊗ σ_{v} , σ_{v} = ρ_{v} × ρ_{v} ,

which is equivariant with respect to the action of the group GSp(4, F) and the group M = Gl(2, F)^{2}/F^{∗}, where M acts through the Weil representation on the left resp. the representation σ_{v} = ρ_{v} × ρ_{v} on the right (ρ_{v} is a cuspidal representation of Gl(2, F_{v})). The image of ζ is one of the two irreducible constituents W p_{±}(ρ_{v}) of the induced representation Ind_{Q}^{G}(1 ⊠ ρ_{v}). See [Wa], lemma 8.2 (2).

We apply these statement for ρ_{v} = σ_{v,2}. We furthermore claim, that the image of ζ is generic, hence is W p_{+}(ρ_{v}). This was expected by Waldspurger. See the remark after lemma 8.2 in [Wa].

Suppose the image of ζ were W p_{−}(σ_{v}). We will now use a result shown in section 11, namely that W p_{−}(σ_{v}) is in the image of the anisotropic theta lift

Θ_{−}(σ_{2,v} × σ_{2,v}) = W p_{−}(σ_{2,v}) .

This implies, that Θ_{+}(σ_{v}) cannot be isomorphic to W p_{−}(σ_{2,v}) either. Otherwise their restriction to the subgroup Sp(4, F_{v}) would have a common irreducible component. By [HPS], thm 9.4 the (big) anisotropic resp. isotropic Weil representations attached to the dual reductive pairs (O_{4}, Sp(4)) and (O(2, 2), Sp(4)) have exactly one common irreducible representation of Sp(4, F_{v}). The prove of the theorem 9.4 of [HPS] furthermore shows, that this common representation is in Howe correspondence with the trivial representation of SO(4, F_{v}). Namely the proof in [HPS] is done by studying Sp(4, F_{v}) invariant distributions on the tensor product of the big Weil representations. The only such distributions factorize over the Dirac distribution of the point zero in the compact Schr”dinger model. The group SO(4, F_{v}) operates therefore trivially on this distribution.

This being said, we again refer to section 11. We will see, that the anisotropic Weil representation with respect to a cuspidal representation σ_{v} = σ_{2,v} × σ_{2,v} does not belong (under restriction to Sp(4, F_{v})) to the trivial representation of SO(4, F_{v}). This could only be the case if σ_{v,2} is a special representation, which we excluded. This completes the proof of lemma 11.

Remeber the temporarily introduced notation (confer the line after lemma 9) to denote π_{+}(σ) any generic irreducible representation π_{i} occuring in the character lift χ(σ_{v}) = ∑ n_{i}π_{i} with positive coefficient n_{i} > 0.

\underline{Theorem 3}: Let σ_{v} be a discrete series representation of the group M_{v}. Then for (any allowed) choice of π_{+}(σ_{v}) we get

π_{+}(σ_{v}) ≅ Θ_{+}(σ_{v}) .

Especially π_{+}(σ_{v}) is uniquely defined.

\underline{Corollary 9}: For cuspidal representations σ_{2,v} of Gl(2, F_{v}) we have for σ_{v} =

σ_{2,v} × σ_{2,v}

π_{+}(σ_{2,v} × σ_{2,v}) = W p_{+}(σ_{2,v}) .

\underline{Proof of thm 3}: For the proof we can assume v to be nonarchimedian. The archimedian case is well known. See [Arthur]. We embed σ_{v} = σ_{2,v} ×σ_{2,v} into some cuspidal irreducible automorphic representation of M(A A). This is always possible, provided we allow some character twists. As our claim is invariant under character twists we can do so. More generally: Every discrete series representation σ_{v} of M_{v} at a finite number of places can be embedded into a global cuspidal representation σ, provided the central characters fit together to a global idele class character. This matching of central characters is irrelevant for us, as the theta lift and also the endoscopic lift commutes with twists by characters. This allows to reduce to a case of torsion characters, where this matching can be forced.

We can therefore furthermore assume, that σ_{∞} = σ_{∞,1} ×σ_{∞,2}, where σ_{∞,i} are holomorphic of weight k_{1} = 2 and k_{2} = k ≥ 3.

Then the representation σ is automatically generic, as non of the two factors σ_{i} can be one dimensional. Furthermore σ_{1} is not isomorphic to σ_{2}. Therefore the global theta lift

Θ_{+}(σ)

is cuspidal, nonvanishing and generic. It is non CAP and a weak lift of σ from M to GSp(4) in our sense. Furthermore it contributes to cohomology as in corollary 3. It is therefore detected by the multiplicity formula of corollary 3 and corollary 5.

On the other hand we have (for any local allowed choices of π_{+}(σ_{v})) the representation

π_{+}(σ) = ⊗′π_{+}(σ_{v})

of GSp(4, A A). Suppose it were automorphic. Then it cannot be a isomorphic locally almost everywhere to an Eisenstein representation, namely at almost all places (e.g. all unramified ones by cor. ) its L-factor is

L_{v}(χ̃_{1}χ̃_{1}^{−1} × χ̃_{2}χ_{1}^{−1} ◁ χ_{1}, s) = L_{v}(σ_{1}, s)L_{v}(σ_{2}) .

Being not of Eisenstein type, π_{+}(σ) would contribute to H_{!}^{3} according to [Lef] and furthermore must be cuspidal, if it is known to be automorphic. All its local components are generic, therefore by theorem 1 the representation π_{+}(σ) it then would be even globally generic. Finally the generic strong multiplicity one theorem (theorem 2) would imply

Θ_{+}(σ) ≅ π_{+}(σ) .

This proves theorem 3, provided π_{+}(σ) is automorphic.

Let us now show, that π_{+}(σ) in fact is automorphic. This is equivalent to

m(π_{+}(σ)) = m(π_{+}(σ_{∞} ⊗ π_{+}(σ)_{fin}) = m_{2}(⊗π_{+}(σ_{v})) > 0 .

The multiplicity formula of corollary 5

m_{2}(π_{+}(σ)_{fin}) = −ε + m_{1}(π_{+}(σ)_{fin}) ≥ −ε

with the constant ε = −1 determined in [W3] implies, that π_{+}(σ) is automorphic. This completes the proof of theorem 3 and furthermore shows m_{1}(π_{+}(σ)_{fin}) = 0.

10. The endoscopic character lift

\underline{Theorem}: For all irreducible unitary discrete series representations σ_{v} = σ_{1,v} ⊗ σ_{2,v} of M we have

χ_{σ_{v}}^{G} = χ_{π_{+}(σ_{v})} − χ_{π_{−}(σ_{v})} ,

where the representations π_{±}(σ_{v}) are as follows:

a) Suppose σ_{v,1} and σ_{v,2} are not isomorphic: Then π_{−}(σ_{v}) is cuspidal. π_{+}(σ_{v}) is cuspidal, if σ_{v} is cuspidal, or is the discrete series representation δ(ρν^{1/2} ◁ χν^{−1/2}) if σ_{v} ≅ Sp ⊗ ρ with cuspidal ρ otherwise.

b) Suppose σ_{1,v} and σ_{2,v} are isomorphic to the same discrete series representation

ρ_{v} of Gl(2, F): Then π_{±}(σ_{v}) = W p_{±}(ρ_{v}).

Furthermore π_{+}(σ_{v}) has a Whittaker model.

\underline{Proof}: Obviously ⟨W_{−}(ρ), W_{−}(ρ)⟩_{G,e} > 0 holds, because W_{±}(ρ) is elliptic. See lemma 6 in section 7. An immediate proof to the conjecture is therefore obtained from the thm 21, p.62 of [SS], which shows that ⟨π_{1}, π_{2}⟩_{G,e} is always an integer for irreducible representations π_{1}, π_{2}. This implies for any elliptic representation W_{+}, W_{−} such that W_{+} + W_{−} is induced ⟨W_{+}, W_{+}⟩_{G,e} = a with a > 0 in ZZ. Then ⟨W_{+}, W_{−}⟩_{G,e} = −a and therefore again ⟨W_{−}, W_{−}⟩_{G,e} = a. This gives ⟨W_{+} − W_{−}, W_{+} − W_{−}⟩_{G,e} = 4a ≥ 4. As a consequence: χ_{σ_{v}}^{G} = χ_{π_{+}} − χ_{π_{−}} if both σ_{i,v} are cuspidal but not isomorphic. This follows because ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩ = 2. Similarily ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩ = 4 if both components are cuspidal and isomorphic to σ_{v_{1}} implies

χ_{σ_{v}}^{G} = χ_{W p_{+}(σ_{v,1})} − χ_{W p_{−}(σ_{v,1})} .

Namely by corollary 9 the expression on the left hand side appears with multiplicity n(σ_{1}) ≠ 0 in the expression

χ_{σ_{v}}^{G} = n(π_{v,1})χ_{π_{v,1}} + n(π_{v,2})χ_{π_{v,2}} + n(π_{v,3})χ_{π_{v,3}} + n(π_{v,4})χ_{π_{v,4}}

+ ∑_{ρ cuspidal} n(ρ)(χ_{W p_{+}(ρ)} − χ_{W p_{−}(ρ)}) ,

of corollary 7. The scalar product of this exceeds ⟨χ_{σ_{v}}^{G}, χ_{σ_{v}}^{G}⟩ = 4 unless n(σ_{1}) = ±1 and shows equality up to a sign. This sign n(σ_{1}) is one. This is seen from the germ expansion, because χ_{σ_{v}} has a Whittaker model and also W p_{+}(σ_{v}) has a Whittaker model, whereas W p_{−}(σ_{v}) does not have a Whittaker model.

This argument includes T_{−} = W p_{−}(Sp), in other words holds also if σ_{v,1} is a special representation. T_{−} = W p_{−}(Sp) is tempered elliptic by lemma 6 and does not admit a Whittaker model because it is an anisotropic theta lift Θ_{−}(1 × 1) = R_{2}(4). (see section 11). Again the argument above shows ⟨T_{+} − T_{−}, T_{+} − T_{−}⟩_{G,e} = 4, hence χ_{Sp×Sp} = T_{+} − T_{−}. Finally T_{+} = Θ_{+}(Sp × Sp) = W p_{+}(Sp) by theorem 3.

\underline{Remark}: We also obtained

⟨W p_{±}(ρ), W p_{±}(ρ)⟩_{G,e} = 1

for all irreducible discrete series representations ρ of Gl(2, F).

\underline{Summary}:

Let F be a p-adic local field. Let M be the group Gl(2, F) × Gl(2, F) and G = GSp(4, F). Let ∗ be the involution of M, which is given by switching the two factors. Let P be the Siegel parabolic in G and let P_{M} be the parabolic in M, which is the product of Gl(2, F) in the first factor and the lower triangular maztrices in the second. Then the

map (x, diag(t_{2}, t_{2})) ↦ blockdiag(t′_{2}det(x)x^{−t}, t_{2}x) allows to identify Levi components of P_{M}

and P. This allows to relate representations

ρ(x, diag(t_{2}, t′_{2})) = σ_{1}(x)χ_{1}(t_{2})χ_{2}(t′_{2})

of the the Levi component of P_{M} to representations

ρ̃(blockdiag(λ · g^{−t}, g)) = σ_{1}(g)χ_{2}(λ/det(g)) .

With these notations and identifications we have constructed a canonical character lift, a homomorphism r(α) = α^{G}

r = r_{M}^{G} : RZZ [M] → RZZ [G]

between the integral Grothendieck groups of finitely generated admissable representations with the following properties:

It describes the endoscopic lift locally

α(f^{M}) = r(α)(f) f ∈ C_{c}^{∞}(G) ,

it commutes with Galois twists (lemma 2)

r(σ^{τ}) = r(σ)^{τ} ,

it commutes with character twists (lemma 1)

r(σ ⊗ χ) = r(σ) ⊗ χ

involution

r(σ^{∗}) = r(σ)

and induction (method 3)

r ∘ r_{P_{M}}^{M} (ρ) = r_{P}^{G}(ρ̃) , ρ ∈ RZZ [Gl(2, F) × F^{∗}]

(ρ̃ is defined via the above mentioned identification of Levi components from ρ) and preserves central characters. Here we used the convention to write r(σ) = r(χ_{σ}), etc. and did not care to distinguish notationally between irreducible representations and their characters.

\underline{Remark}: The group generated by M = GSO(2, 2) and the outer automorphism ∗ is isomorphic to the group GO(2, 2), the group of orthogonal smilitudes of a split quadratic form of rank 4. In this disguise the endoscopic lift r is related to the theta lift from

GO(2, 2) to G.

Let us explicitely describe the lift r on the irreducible tempered representations of M. Irreducible admissable representations σ of G appear in the form σ = σ_{1} ×σ_{2}, where σ_{i} are irreducible admissable representations of Gl(2, F) with equal central character ω_{σ_{i}}. The relevant cases are:

\underline{1. Discrete series case}: If σ is irreducible discrete series representation of M, then

r(σ) = π_{+}(σ) − π_{−}(σ) ,

where π_{+}(σ) and π_{−}(σ) are irreducible admissable representations of G. π_{+}(σ) = Θ_{+}(σ) has a Whittaker model. The representations π_{±}(σ) are descibed as follows:

1a) \underline{If σ_{i} are both cuspidal and not isomorphic}, then π_{±}(σ) are both cuspidal.

Now assume still σ is a discrete series representation of M over the local field, but not of the form already discussed. There are three subcases 1b, 1c and 1d to be considered.

1b) \underline{The case, where σ_{1} ≅ σ_{2} and σ_{1} is cuspidal}. In this case we have already indicated that π_{±}(σ) = W p_{±}(σ_{1}). Remember that these representations are elliptic relatives in the sense of Kazhdan, and especially are not cuspidal. Only one of them has a Whittaker model. In section we show

π_{−}(σ) = W p_{−}(σ_{1}) = Θ_{−}(σ_{1}) .

Note that Θ_{−}(σ_{1}) is an anisotropic Theta lift, which is in general welldefined only up to some additional hypothesis (e.g. Howe conjecture, residue characteristic

≠ 2).

r(σ_{1} × σ_{1}) = W p_{+}(σ_{1}) − W p_{−}(σ_{1}) = Θ_{+}(σ_{1}) − Θ_{−}(σ_{1}) .

1c) \underline{If σ_{1} is cuspidal, and σ_{2} = Sp(χ) is the special representation}, then we have seen π_{+}(σ) = δ(σ_{1}ν^{1/2}◁χν^{−1/2}) is discrete series, but not cuspidal and π_{−}(σ) is cuspidal.

r(σ_{1} × Sp) = δ(σ_{1}ν^{1/2} ◁ χν^{−1/2}) − π_{−}(σ_{1}) .

1d) \underline{If σ_{i} = Sp(χ) are both special}, then π_{±}(σ) = W p_{±}(σ_{1}) = (T_{+} − T_{−}) ⊗ χ are tem-

pered. The discussion for the nonregular case b) in section 6 gave T_{−} = R_{2}(4), which is the anisotropic theta lift Θ_{−}(1 × 1). Therefore T_{−} has no Whittaker model (see section 11). T_{+} = Θ_{+}(σ) is the indefinite theta lift Θ_{+}(σ).

r(Sp × Sp) = T_{+} − T_{−} = Θ_{+}(Sp × Sp) − Θ_{−}(1 × 1) .

This completes the description of the lifts r(σ) for discrete series representations. Let us now consider the tempered case 2 and two further cases, which are important for the Saito-Kurokawa lift. We state the results and after that, give the proof.

2. \underline{Tempered non discrete series}: Suppose that σ = σ_{1} × σ_{2} is such that σ_{1} is irreducible tempered on Gl(2, F) and that σ_{2} = Ind(χ̃_{1}, χ̃_{2}) is induced from a pair of unitary characters. Then r(σ) is irreducible and tempered.

r(σ_{1} × Ind(χ̃_{1}, χ̃_{2})) = σ_{1}χ̃_{1}^{−1} ◁ χ̃_{1} .

Next we describe the degenerate cases, where σ_{2} = 1 is the trivial representation. This should give the representations appearing in the Saito Kurokawa lift (Arthur packets). This the nonarchimedian analog of [A2], case 1.4.1 with f_{M}(ψ) = r(σ_{1} ×1) in the notation of loc. cit.

3. \underline{Degenerate case}: If σ_{1} is cuspidal (ω_{σ_{1}} = 1), then

r(σ_{1} × 1) = J_{P} (ν^{1/2}, σ_{1} ⊠ 1) + π_{−}(σ_{1})

is the sum of two irreducible representations. The first is nontempered, the second is cuspidal . The second representation π_{−}(σ_{1}) also appears in the lift r(σ_{1} × Sp). We have

r(Sp × 1) = J′ + T_{−}

is the sum of two irreducible representations, where T_{−} is the tempered representation already appearing in the lift r(Sp × Sp). Finally

r(σ_{1} × 1) = χ_{1} ∘ det ◁ χ_{2}

is irreducible nontempered, if σ_{1} = Ind(χ_{1}, χ_{2}) is tempered.

Last not least

4. \underline{The totally degenerate case}: r(1 × 1) = J − J′.

\underline{Proof}: All statements except the explicit description of the lift r in the cases 2.,3. and 4. were already gathered. The remaining statement are then easily deduced. For the conveniance of the reader, let me give the arguments.

Observe the \underline{induction formula}

r(Ind(χ̃_{1}, χ̃_{2}) × Ind(χ_{1}, χ_{2})) = χ̃_{1}χ_{1}^{−1} × χ̃_{2}χ_{1}^{−1} ◁ χ_{1} ,

more generally

r(σ_{1} × Ind(χ_{1}, χ_{2})) = σ_{1}χ_{1}^{−1} ◁ χ_{1} .

This is just the compatibility of the lift r with induction. Namely ρ(x, diag(t_{2}, t′_{2})) =

σ_{1}(x)χ_{1}(t_{2})χ_{2}(t′_{2}) relates to ρ̃(blockdiag(λg^{−t}, g)) = σ_{1}(g)χ_{2}(λ/det(g)) under the comparison

map of the Levi components of P_{M} and P. Then comparison map is

(x, diag(t_{2}, t′_{2})) ∈ P_{M} ↦ blockdiag(λg^{−t}, g) = blockdiag(t′_{2}det(x)x^{−t}, t_{2}x) ∈ P , t′_{2} = 1 .

Using our previous conventions on induction from P to G used in section 6 we switch notation once more and write blockdiag(λ · g^{−t}, g) = blockdiag(x, τ · x^{−t}). This implies g =

τ · x^{−t} and λ = τ. Under ρ̃ the element blockdiag(x, τx^{−t} is mapped to σ_{1}(g)χ_{2}(λ/det(g)) = (σ_{1}χ_{2}^{−1})(g)χ_{2}(λ) = (σ_{1}χ_{2}^{−1})(τx^{−t})χ_{2}(τ) = (σ_{1}χ_{2}^{−1})^{∗}(x)(ω_{σ_{1}}χ_{2}^{−2}χ_{2})(τ) = (σ_{1}χ_{1}^{−1})(x)χ_{1}(τ). This

implies, that the induced reprentations is σ_{1}χ_{1}^{−1} ◁ χ_{1} according to our notation in section 6. This proves the induction formula.

\underline{Proof of case 2}: This being said it is clear, that r(Ind(χ̃_{1}, χ̃_{2}) × Ind(χ_{1}, χ_{2})) is irreducible tempered for unitary characters χ_{i}, χ̃_{i}, because the unitary principal series on G are irreducible by the results of Tadic and Rodier. If σ_{1} is special or cuspidal and χ_{1} is unitary, then σ_{1}χ_{1}^{−1} ◁ χ_{1} is irreducible tempered. This follows from the results of Shahidi (when σ_{1} is cuspidal) respectively by the results of Tadic (when σ_{1} = Sp). In the case of the special representation see the discussion preceeding lemma 7 in section 7. One has to superficially distinguish the subcases, where χ_{1}^{2} = 1 or not! This completes the proof of case 2.

\underline{Proof of case 3}: Let us turn to the degenerate case, where σ_{2} = 1 is the trivial representation. First assume that σ_{1} is cuspidal and without restriction of generality with trivial central character. Then

r(σ) = r(σ_{1} × Ind(ν^{−1/2}, ν^{1/2})) − r(σ_{1} × Sp)
=(δ(σ_{1}ν^{1/2} ◁ ν^{−1/2}) + J_{P} (σ_{1}ν^{1/2} ◁ ν^{−1/2}))−(δ(σ_{1}ν^{1/2} ◁ ν^{−1/2}) − π_{−}(σ_{1}))
= J_{P} (σ_{1}ν^{1/2} ◁ ν^{−1/2}) + π_{−}(σ_{1}) .

The first term is a non tempered Langlands quotient, which arises together with the discrete series representation D = δ(σ_{1}ν^{1/2} ◁ ν^{−1/2}). The second term π_{−}(σ_{1}) is a cuspidal representation. See the discussion of the results of Shahidi concerning cuspidal induction from P.

Next consider the case r(Sp × 1). We have r(Sp × 1) = r(Sp × Ind(ν^{−1/2}, ν^{1/2})) − r(Sp × Sp).

But r(Sp × Ind(ν^{−1/2}, ν^{1/2})) = Spν^{1/2} ◁ ν^{−1/2}. This we denoted L in our discussion of the nonregular case a) and we have shown there L = T_{+} + J′. Remember, J′ = J_{P} (ν^{1/2}, Sp ⊠ ν^{−1/2}) is a irreducible nontempered Langlands quotient. On the other hand r(Sp × Sp) =

T_{+} − T_{−}, therefore r(Sp × 1) = J′ + T_{−}.

It remains to consider r(Ind(χ̃_{1}, χ̃_{2})×1) for unitary characters χ̃_{i}. We then have automatically χ̃_{1}χ̃_{2} = 1. In the Grothendieck group Sp + 1 = Ind(ν^{−1/2}, ν^{1/2}), thus r(Ind(χ̃_{1}, χ̃_{2}) × 1) is in the Grothendieck group equal to χ̃_{1}ν^{1/2} × χ̃_{2}ν^{1/2} ◁ ν^{−1/2} − r(Ind(χ̃_{1}, χ̃_{2}) × Sp) using the induction formula. Furthermore r(Ind(χ̃_{1}, χ̃_{2}) × Sp) = r(Sp × Ind(χ̃_{1}, χ̃_{2})) = Spχ̃_{1}^{−1} ◁ χ̃_{1} once more by the induction formula. We used in addition that Ind(χ̃_{1}, χ̃_{2}) does not depend on the choice of the Borel group, when considered in the Grothendieck group. A similar discussion gives on the other hand χ̃_{1}ν^{1/2} × χ̃_{2}ν^{1/2} ◁ ν^{−1/2} = χ̃_{1}ν^{1/2} × χ̃_{1}ν^{−1/2} ◁ χ̃_{2}, again viewer in the Grothendieck group. In section 7 it was shows, that the latter has exactly two irreducible constituents. One of it is Sp(χ̃_{1}) ◁ χ̃_{2}, the other is χ̃_{1} ∘ det ◁ χ̃_{2}, which is non tempered. With this we get r(Ind(χ̃_{1}, χ̃_{2}) × 1) = χ̃_{1} ∘ det ◁ χ̃_{2}.

\underline{Proof of case 4}: Finally we compute r(1×1). Consider r(Ind(ν^{1/2}, ν^{−1/2})×Ind(ν^{−1/2}, ν^{1/2})). By the induction formula this is ν × 1 ◁ ν^{−1/2} = 1 × ν ◁ ν^{−1/2} (viewed in the Grothendieck group). This is J + J′ + T_{+} + T_{−} according to our discussion of case b). However we

already know r(Sp × Sp) = T_{+} − T_{−}. Therefore 2 · r(Sp × 1) + r(1 × 1) = J + J′ + 2T_{−}. Substract 2r(Sp × 1) = 2J′ + 2T_{−}. This gives r(1 × 1) = J − J′.

\underline{Final remark}: By imbedding into a totally real global field K, the the local description of the endoscopic character lift carries over to arbitrary local fields of characteristic zero once an analog of corollary 1 is shown for the Shimura variety attached to Res_{K/Q}(GSp(4)). This should be an easy excersise to do, however the author did not check all necessary details and leaves this to the reader.

\underline{11. The Anisotropic Theta lift}:

Consider the group M_{c} = D^{∗} × D^{∗}/{(x, x^{−1}) | x ∈ F^{∗}} ↪ GO(4, q), where D is a quaternion skew field over F = Q_{p} and where

q = Norm_{D/F} : D → F

is the norm form v\overline{v} = q(v). The group M_{c} acts on D via

(d_{1}, d_{2}) · v = d_{1}v\overline{d}_{2} , d_{1}, d_{2} ∈ D^{∗}, v ∈ D .

Let denote SO the kernel of the map M_{c} → F^{∗}, which maps (d_{1}, d_{2}) to Norm_{D/F} (d_{1}d_{2}). The group SO is compact. Irreducible admissable representations of M_{c} are of the form σ̌(d_{1}, d_{2}) = σ̌(d_{1}) ⊗ σ̌_{2}(d_{2}), where σ̌_{i} are irreducible admissable with coinciding central character and vice versa.

Put 2 · B(v, w) = q(v + w) − q(v) − q(w). Then 2 · B(v, w) = v\overline{w} + w\overline{v} or equivalently B(v, w) =

\frac{1}{2}tr_{D/F} (v\overline{w}). Extend the bilinear form B on D to B ⊕B on D^{2}. Fix some nontrival additive character ψ of F. Let S(D) and S(D^{2}) be the space of Schwartz-Bruhat functions on the vector spaces D resp D^{2}. We also consider the natural action of SO on S(D) and S(D^{2}). Put

Q(T; X, Y) = T race(T · ( \begin{matrix}B(x_{1}, y_{1}) & B(x_{1}, y_{2}) \\ B(x_{2}, y_{1}) & B(x_{2}, y_{2}) \end{matrix} )) , X = (x_{1}, x_{2}), Y = (y_{1}, y_{2}) .

Choose some nontrivial character ψ_{1} of F. Then every nontrivial additive character ψ(x) is of the form ψ_{t}(x) = ψ_{1}(tx) for some t ∈ F^{∗}.

a) Consider the \underline{Weil representation} π of GSp(4, F) on S(D^{2} × F^{∗}) defined by

π(( \begin{matrix}E & T \\ 0 & E \end{matrix} ))Φ(X, t) = ψ_{t}(Q(T, X, X))Φ(X, t) ,

where X ∈ D^{2}.

π(( \begin{matrix}A & 0 \\ 0 & A^{−t} \end{matrix} ))Φ(X, t) = |det(A)|^{2}Φ(X ∘ A, t) ,

where X ∘ A = (x_{1}a_{11} + x_{2}a_{21}, x_{1}a_{12} + x_{2}a_{22}) and A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ). Finally

π(( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ))Φ(X, t) = ε_{2}(q, ψ_{t})Φ̂(X, t) ,

where

Φ̂(X, t) = ∫_{D} Φ(Y, t)ψ_{t}(2Q(E, X, Y))dY_{ψ_{t}} .

Here dY_{ψ_{t}} is the selfdual Haar measure on D with respect to the Fourier transform defined via ψ_{t}.

π(( \begin{matrix}E & 0 \\ 0 & λE \end{matrix} ) Φ(X, t) = Φ(X, tλ^{−1}) .

Then π is a unitary representation on S(D^{2} × F^{∗}) ⊂ L^{2}(D^{2} × F^{∗}). This action commutes with the unitary action of M_{c} defined by

π((d_{1}, d_{2}))Φ(X, t) = |q(d_{1}d_{2})|^{2}Φ(d_{1}X\overline{d}_{2}, q(d_{1}d_{2})^{−1}t) .

Note that λ ∈ F^{∗} viewed as element of the center {(λ, 1)} of M_{c} or as element diag(λE, λE) acts by the same assignment Φ(X, t) ↦ |λ|^{4}Φ(λX, t/λ^{2}).

We have a surjective map

S(D^{2} × F^{∗}) → S(D^{2} × F^{∗}, ω)
Φ(X, t) ↦ ∫_{F^{∗}} |λ|^{4}Φ(λX, tλ^{−2})ω(λ)^{−1}d^{•}λ

for a quasicharacter ω of F^{∗}. Every equivariant map from S(D^{2} × F^{∗}) to some vector space, on which the center of G or M_{c} acts by ω factors through this map. There is an induced action of G × M_{c} on S(D^{2} × F^{∗}, ω), which makes the map equivariant with respect to the action of G × M_{c}.

Let denote G^{0} resp. M_{c}^{0} the subgroups of G resp. M_{c} (of finite index) of all elements with similitude factor resp. norm in (F^{∗})^{2}. Let S_{t_{0}}(D^{2} × F^{∗}, ω) denote the subspace of S(D^{2} × F^{∗}, ω) of all functions with support with respect to the second variable in t_{0}(F^{∗})^{2}. Then by choosing representatives t_{0} we get

S(D^{2} × F^{∗}, ω) = ⨁_{t_{0}∈F^{∗}/(F^{∗})^{2}} S_{t_{0}}(D^{2} × F^{∗}, ω) .

The subspaces S_{t_{0}}(D^{2} × F^{∗}, ω) are preserved by the action of G^{0} × M_{c}^{0}. Let ω_{0} be the restriction of ω to the subgroup {±1} ⊂ F^{∗}. We identify

ev_{t_{0}} : S_{t_{0}}(D^{2} × F^{∗}, ω) ≅ S(D^{2}, ω_{0})
Φ(X, t) ↦ Φ(X, t_{0}) .

Using this identification we can transport the action of G^{0} × M_{c}^{0} to S(D^{2}). This action will be denoted π_{t_{0}}. Then as a G^{0} × M_{c}^{0} module we get

S(D^{2} × F^{∗}, ω) ≅ ⨁_{t_{0}∈F^{∗}/(F^{∗})^{2}}(S(D^{2}), π_{t_{0}}) .

\underline{Corollary 10}: The representation π̃ on S(D^{2} × F^{∗}, ω) is admissable. Its restriction to the subgroup Sp(4, F) is preunitary.

\underline{Proof}: It is enough for both statements to consider the action of the subgroup Sp(4, F). One is thus reduced to the finitely many representations S(D^{2}), π_{t_{0}}), for which the claim is easily verified.

b) Similarily one defines a unitary Weil representation π̃ of Gl(2, F) on S(D × F^{∗}) by the formulas

π̃(( \begin{matrix}1 & τ \\ 0 & 1 \end{matrix} ))Φ(x, t) = ψ_{t}(τB(x, x))Φ(x, t) ,

π̃(( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ))Φ(x, t) = |a|^{2}Φ(xa, t) ,

π̃(( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ))Φ(x, t) = ε_{1}(q, ψ_{t})Φ̂(x, t) ,

where ε_{1}(q, ψ_{1}) = −1 if ψ_{1} is the standard additive character of F and dy_{ψ_{t}} is the selfdual Haar measure on D for Fourier transform with respect to ψ_{t}.

Φ̂(x, t) = ∫_{D} Φ(y, t)ψ_{t}(2 · B(x, y))dy_{ψ_{t}} .

Again

π̃(( \begin{matrix}1 & 0 \\ 0 & λ \end{matrix} ) Φ(x, t) = Φ(x, tλ^{−1}) .

This defines a unitary representation of Gl(2, F). It commutes with the unitary action of M_{c} defined by

π̃((d_{1}, d_{2}))Φ(x, t) = |q(d_{1}d_{2})|Φ(d_{1}x\overline{d}_{2}, q(d_{1}d_{2})^{−1}t) .

Similar as above one can restrict to a subgroup of elements with determinant in (F^{∗})^{2} and descibe the resulting representations within S(D).

\underline{Global situation}:

Let us switch notation and let F be a number field with adele ring A A and D a global quaternion algebra over F. Then there are global analogs π and π̃ acting on S(D_{A A}^{2} × A A^{∗}) resp S(D_{A A} × A A^{∗}). The theta distribution

ϑ : S(D_{A A}^{i} × A A^{∗}) → C , (i = 1, 2)

ϑ_{Φ} = ∑_{(X,t)∈D^{i}×F^{∗}} Φ(X, t)

are welldefined. Then

ϑ_{Φ}(m, g) = ϑ_{π(m,g)Φ} , (m, g) ∈ M_{c}(A A) × G(A A)

is of moderate growth on (M_{c} × G)(A A)/(M_{c} × G)(F). For an irreducible subspace τ of the space of cuspforms on M_{c}(A A)/M_{c}(F) the theta lift Θ_{−}(τ) is the G(A A) invariant space of automorphic forms on G = GSp(4, A A) spanned by all functions

∫_{M_{c}(A A )/M_{c}(F)} ϑ_{Φ}(m × g)\overline{f}(m)dm ,

Φ runs over S(D_{A A}^{2} × A A^{∗}) and f runs over all functions in the representation space of τ. Let ω be the central character of τ. Suppose Θ_{−}(τ) is contained in the space of cuspforms and let π(τ) some irreducible constituent of it. By evaluating with Φ = Φ_{p}Φ^{P} one gets local maps at the prime p

ϑ_{Φ^{p}} : S(D^{2} × F^{∗}) → π_{p} .

They factorize over

ϑ_{Φ^{p}} : S(D^{2} × F^{∗}, ω_{p}) → π_{p} .

\underline{The anisotropic theta lift Θ_{−}(local theory)}:

Consider the maximal quotient W of S(D × F^{∗}) transforming with respect to a fixed irreducible representation σ̌^{∗} of the group M_{c}. Let ω be the central character of σ̌. Then W is a quotient of S(D × F^{∗}, ω). Then W as a module under GSp(4, F) can be embedded into a finite direct sum of copies of the following GSp(4, F) module

S(D × F^{∗}, σ̌) = {Φ ∈ S(D × F^{∗}, ω) ⊗ Vσ̌ | π̃(h)Φ = σ̌(h)Φ , h ∈ M_{c}} .
π̃(d_{1}, d_{2})Φ(x, t) = |q(d_{1}, d_{2})|Φ(d_{1}x\overline{d}_{2}, t/q(d_{1}d_{2})) .

This uses that M_{c} is compact mod the center. This also implies that the representations σ̌ are finite dimensional. Similarily define the spaces S(D^{2} × F^{∗}, σ̌).

\underline{Lemma}: S(D × F^{∗}, σ̌) = 0 unless σ̌ ≅ σ̌_{1} × σ̌_{2} with σ̌_{1} ≅ σ̌_{2}.

\underline{Proof}: The action of M_{c} commutes with the natural action of F^{∗} on S(D × F^{∗}). If Φ(x, t) ∈ S(D × F^{∗}, σ̌) is not zero, then there exists a quasicharacter χ of F^{∗}, such

that Φ(x) = ∫_{z∈F^{∗}} ∫_{F^{∗}} Φ′(zx, tz^{−2})|z|ω(z)^{−1}χ(t)^{−1}d^{•}td^{•}z ≠ 0. Here Φ′ is some lift of Φ in S(D^{2} × F^{∗}) ⊗ Vσ̌. We have Φ ∈ S(D) and Φ(d_{1}x\overline{d}_{2}) ≅ (χν)(q(d_{1}d_{2}))σ̌(d_{1}, d_{2})Φ(x). Note that

in the following we will use the isomorphism

σ̌_{2}(d_{2}^{∗}) ≅ σ̌_{2}^{∗}(d_{2}) , d_{2}^{∗} = \overline{d}_{2}^{−1} = d_{2}/q(d_{2}) .

Then Φ(d_{1}xd_{2}^{−1})(χν)(q(d_{1}/d_{2}))σ̌_{1}(d_{1})σ̌_{2}^{∗}(d_{2})Φ(x) and the Peter-Weyl theorem as in [HPS], thm. 9.1 implies that σ̌_{1} ≅ σ̌_{2}.

\underline{Lemma}: For all σ̌ the space S(D^{2} × F^{∗}, σ̌) is nontrivial.

\underline{Proof}: As above this is reduced to [HPS], thm 9.1.

\underline{Definition}: Let Θ_{−}(σ̌) denote (any) irreducible admissable quotient representation of the

GSp(4, F) module (S(D^{2} × F^{∗}, σ̌), π(σ̌)).

\underline{Remark}: The Howe conjecture would imply uniqueness of Θ_{−}(σ̌).

\underline{Lemma}: The representation π̃(σ̌) of Gl(2, F) on S(D × F^{∗}, σ̌) is irreducible and isomorphic to the Jaquet-Langlands lift σ = JL(σ̌_{2}) if σ̌ ≅ σ̌_{2} × σ̌_{2}.

\underline{Proof}: The Jaquet-Langlands lift JL(σ̌) of σ̌ is realized as follows: Let ω be the central character of σ̌_{2}. Let D^{1} be the norm 1 subgroup of D^{∗}. Define a representation ρ of Gl(2, F) on the space of functions Φ(y) in

S(D, σ̌_{2}) = {Φ ∈ S(D) × Vσ̌_{2} | Φ(xh) = σ̌_{2}(h)^{−1}Φ(x) , h ∈ D^{1}}

by

ρ(( \begin{matrix}1 & τ \\ 0 & 1 \end{matrix} ))Φ(y) = ψ(τB(x, x))Φ(y) ,

ρ(( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ))Φ(y) = |a|^{2}Φ(ya) ,

ρ(( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ))Φ(y) = ε_{1}(q, ψ)Φ̂(y) ,

where ε_{1}(q, ψ) = −1 if ψ = ψ_{1} is the standard additive character of F and dy_{ψ} is the selfdual Haar measure on D for Fourier transform with respect to ψ.

Φ̂(y) = ∫_{D} Φ(z)ψ_{t}(2 · B(z, y))dz_{ψ} .

Finally

ρ(( \begin{matrix}λ & 0 \\ 0 & 1 \end{matrix} ) Φ(y) = |q(x_{λ})|σ̌_{2}(x_{λ})Φ(yx_{λ}) ,

where x_{λ} ∈ D^{1} \ D^{∗} is any element with norm q(x_{λ}) = λ. Especially

ρ(( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} ) Φ(y) = ω(λ)Φ(y) .

It is well known, that ρ is a direct sum of dim(σ̌_{2}) copies of the Jaquet-Langlands lift JL(σ̌_{2}). See [ ] (use the Kirillov model).

\underline{The intertwining map b}:

Let V ∗ ; σ̌_{1} be the dual space of Vσ̌_{1}. Then we define a map

b : V ∗ ; σ̌_{1} ⊗ S(D × F^{∗}, σ̌_{1} × σ̌_{2}) ⟶ S(D, σ̌_{2})

by

v_{1}^{∗} ⊗ Φ(y, t) ↦ v_{1}^{∗}(Φ(y, 1)) .

The map b is \underline{welldefined}: Namely for h ∈ D^{1} we have h^{∗} = h and q(h) = 1. Therefore

b(v_{1}^{∗} ⊗ Φ)(yh) = v_{1}^{∗}(Φ(yh, 1)) = v_{1}^{∗}(|q(h^{−1})|Φ(y\overline{h}^{−1}, q(h)))
= v_{1}^{∗}(π̃(1, h^{−1})Φ(y, 1)) = v_{1}^{∗}(σ̌_{2}(h^{−1})ϕ(y, 1)) = σ̌_{2}(h^{−1})v_{1}^{∗}(Φ(y, 1))
= σ̌_{2}(h)^{−1}b(v_{1}^{∗} ⊗ Φ)(y) .

This implies b(v_{1}^{∗} ⊗ Φ) ∈ S(D, σ̌_{2}).

The map b is \underline{equivariant} with respect to the representations 1_{V ∗ ; σ̌_{1}} ⊗ π̃ and ρ of the group Gl(2, F). As b is essentially defined by the specialisation t = 1, this assertion is clear for the action of the subgroup Sl(2, F). It is therefore enough to compare

b(v_{1}^{∗} ⊗ π̃(diag(λ, 1))Φ(y, t)) = v_{1}^{∗}(ω(λ)Φ(y, λ))

and

ρ(diag(λ, 1))b(v_{1}^{∗} ⊗ Φ(y, t)) = |λ|σ̌_{2}(x_{λ})v_{1}^{∗}(Φ(yx_{λ}, 1)) , q(x_{λ}) = λ

= |λ|σ̌_{2}(x_{λ})v_{1}^{∗}(Φ(yx_{λ}, λ/q(x_{λ}))
= σ̌_{2}(x_{λ})v_{1}^{∗}(σ̌_{2}(\overline{x}_{λ})Φ(y, λ))
= σ̌_{2}(x_{λ}\overline{x}_{λ})v_{1}^{∗}(Φ(y, λ))
= v_{1}^{∗}(ω(λ)Φ(y, λ)) .

The map b is \underline{injective}: For that we can assume σ̌ = σ̌_{1} × σ̌_{2} with σ_{1} ≅ σ_{2}. Otherwise the space under consideration is zero. We can especially assume Vσ̌_{1} = C^{d} and σ̌_{2}(d) = σ̌_{1}(d_{2}^{∗})^{−t} (acting on C^{d}). Then Vσ̌ is identified with the space of matrices M_{d,d}(C)

σ̌(d_{1}, d_{2})X = σ̌_{1}(d_{1})Xσ̌_{1}(\overline{d}_{2}) , Vσ̌ = M_{d,d}(C) .

With this assumptions we see that for Φ(x, t) ∈ S(D × F^{∗}, σ̌) and d ≠ 0 we have

|q(d)|ϕ(d, t/q(d)) = σ̌(d, 1)Φ(1, t) = σ̌(1, \overline{d})Φ(1, t) .

Thus

σ̌_{1}(d)Φ(1, t) = Φ(1, t)σ̌_{1}(d)

for all d ∈ D^{∗}, hence

Φ(1, t) = φ(1, t) · id , φ(1, t) ∈ C .

We see that for Φ(x, t) ∈ S(D × F^{∗}, σ̌) and x ≠ 0

Φ(x, t) = φ(1, tq(x))|q(x)|^{−1} · σ̌_{1}(x) .

If dim(σ_{2}) > 1 then Φ(0, t) = 0 automatically. Therefore we have proven one direction of the

\underline{Claim}: If d = dim(σ_{2}) > 1 and σ̌ acting on M_{d,d}(C) as above, then

S(D × F^{∗}, σ̌) = {Φ(x, t) = |q(x)|^{−1}σ̌_{1}(x)φ(tq(x)) | φ ∈ S(F^{∗})} ≅ S(F^{∗}) ,

understood in the sense that Φ(0, t) = 0. If dim(σ̌_{2}) = 1 then S(D ×F^{∗}, σ̌) has codimension 1 in the right side.

The other direction of the claim is obvious.

Let e_{i}^{∗} be a basis of V ∗ ; σ̌_{1}. From the claim follows

b(∑_{i=1}^{d} e_{i}^{∗} ⊗ Φ_{i}(x, t)) = |q(y)|^{−1} ∑_{i} φ_{i}(q(y))e_{i}^{∗}(σ_{1}(x)) ∈ C^{d} .

However the functions e_{i}^{∗}(σ_{1}(x)) are all linear independent and nowhere vanishing on D^{∗}. Therefore vanishing of this expression implies Φ_{i}(x, t) = 0 for all x ≠ 0, hence Φ_{i}(x, t) = 0. Therefore the map b is injective. This proves that the representation π̃ on S(D × F^{∗}, σ̌) is irreducible and isomorphic to JL(σ̌_{2}). Hence the lemma is proven. Using Kirillov models one finds furthermore

\underline{Corollary 11}: b is an isomorphism.

\underline{Jaquet modules of the anisotropic Weil representation Θ_{−}}:

We considered two Weil representations above, namely the representation π of GSp(4, F) and the representation π̃ of Gl(2, F). They are related as follows. Consider the maximal parabolic subgroup Q of GSp(4, F), then π̃ is the representation which arises in the normalized Jaquet module with respect to the parabolic Q. For this consider the subgroup N ⊂ Q consisting of all matrices

n(τ) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & τ \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , τ ∈ F .

N is the center of the unipotent radical of Q. Let Φ ∈ S(D^{2} × F^{∗}). Then

(π(n(τ)) − 1)Φ(x_{1}, x_{2}, t) = (ψ_{t}(τB(x_{2}, x_{2})) − 1)Φ(x_{1}, x_{2}, t) .

Hence

\overline{ψ} : S(D^{2} × F^{∗})/(π(N) − 1)S(D^{2} × F^{∗}) ≅ S(D × F^{∗}) ⊗ ν .

This isomorphism \overline{ψ} is induced by the Q×M_{c} equivariant map defined by ψ : Φ(x_{1}, x_{2}, t) ↦ Φ(x_{1}, 0, t) (hopefully not to be confused with the additive character). Namely every Schwartz-Bruhat function with Φ(x_{1}, 0, t) = 0 is a finite linear combination of products ϕ_{1}(x_{1}, t)ϕ_{2}(x_{2}) with ϕ_{1} ∈ S(D×F^{∗}) and ϕ_{2} ∈ S(D) and ϕ_{2}(0) = 0. So one is reduced to show that ϕ_{2}(0) = 0 implies that there exists ϕ_{3}(x_{2}, t) ∈ S(D ×F^{∗}) and τ ∈ F^{∗} such that ϕ_{2}(x_{2}) =

(ψ_{t}(τB(x_{2}, x_{2})) − 1)ϕ_{3}(x_{2}, t) for all (t, x_{2} in the support Supp(ϕ_{1}) × Supp(ϕ_{2}) ⊂ F^{∗} × D^{∗}. This

is obviously possible. Furthermore it is clear that

ψ ∈ Hom_{Q×M_{c}}(δ_{Q}^{−1/2} ⊗ S(D^{2} × F^{∗}), S(D × F^{∗}) ⊗ ν)

with Q equivariance given as follows. For η ∈ Q

η = ( \begin{matrix}α & 0 & β & ∗ \\ ∗ & a & ∗ & ∗ \\ γ & 0 & δ & ∗ \\ 0 & 0 & 0 & λa^{−1} \end{matrix} )

we get

(π(η)Φ)(x, 0, t) = |a|^{2}(π̃(( \begin{matrix}α & β \\ γ & δ \end{matrix} ))Φ(y, 0, t))_{y=x} .

Especially the unipotent radical of Q operates trivially. We therefore get

S(D^{2} × F^{∗})/(π(Rad(Q)) − 1) ≅ S(D^{2} × F^{∗})/(π(N) − 1) ≅ S(D × F^{∗}) .

Note that δ_{Q}^{1/2}(η) = |a|^{2}|λ|^{−1}. Therefore the normalized Jaquet module of π with respect to Q is given by

\overline{ψ} : δ_{Q}^{−1/2}(S(D^{2} × F^{∗}), π)(Q) ≅ 1 ⊠ (S(D × F^{∗}), π̃ ⊗ ν) .

Here the Levi component of Q is viewed as the group F^{∗} × Gl(2, F) using the matrix coordinates a and α, β, γ, δ of η as above.

By passage to the quotients with central ω-action by central integration we also obtain

\overline{ψ}(ω) : δ_{Q}^{−1/2}(S(D^{2} × F^{∗}, ω), π)(Q) ≅ 1 ⊠ (S(D × F^{∗}, ν^{−1}ω), π̃ ⊗ ν) .

and by splitting up into a direct sum of maps we obtain from that

\overline{ψ}(σ̌) : δ_{Q}^{−1/2}(S(D^{2} × F^{∗}, σ̌), π(σ̌))(Q) ≅ 1 ⊠ (S(D × F^{∗}, ν^{−1} ⊗ σ̌), π̃(ν^{−1} ⊗ σ̌) ⊗ ν) .

Note that

(S(D × F^{∗}, ν^{−1} ⊗ σ̌), π̃(ν^{−1} ⊗ σ̌) ⊗ ν) ≅ (S(D × F^{∗}, σ̌), π̃(σ̌)) ,

because the Jaquet-Langlands lift is compatible with character twists.

The map ψ(σ̌) induces by adjunction (e.g. Silberger Thm. 2.4.3)

Hom_{G}((S(D^{2} × F^{∗}), π(σ̌)), Ind_{Q}^{G}(τ)) ≅ Hom_{Q}(δ_{Q}^{−1/2}π(Q), τ) ,

(for all admissable representations τ of Q/Rad(Q)) a nontrivial map

Ψ(σ̌) : (S(D^{2} × F^{∗}, σ̌), π(σ̌)) → Ind_{Q}^{G}(1 ⊠ (S(D × F^{∗}, σ̌), π̃(σ̌)) ,

which is the universal map into Q/Rad(Q)-induced representations of G.

\underline{Whittaker models}: We claim that (S(D^{2}), π) does not admit a nontrivial Whittaker functional. Let ψ_{1}, ψ_{2} be nontrivial additive characters. Suppose L were a Whittaker functional, i.e.

L : S(D^{2}) → C , L(π(n(x, y, u, v, t)Φ) = ψ_{1}(x)ψ_{2}(y)L(Φ)

for all Φ ∈ S(D^{2}) and all

n(x, y, u, v, t) = ( \begin{matrix}1 & 0 & x & u \\ y & 1 & v & t \\ 0 & 0 & 1 & −y \\ 0 & 0 & 0 & 1 \end{matrix} ) ∈ Q

in the unipotent radical U of the Borel subgroup. The commutator group consists of the matrices with x = y = 0.

Then L(π(n(t))Φ) = L(Φ) holds for all t ∈ F. Hence L = \overline{L} ∘ ψ factorizes over a functional \overline{L} : S(D) → C. Abbreviate ψ(Φ) = \overline{Φ} ∈ S(D). Then by Q equivariance

ψ_{2}(y)\overline{L}(\overline{Φ}) = ψ_{2}(y)L(Φ) = L(π(n(0, y, ∗, ∗, ∗)Φ) = \overline{L}(\overline{Φ})

for all y ∈ F. Therefore \overline{L}(\overline{Φ}) = 0 for all \overline{Φ}, hence L = 0.

\underline{Lemma}: The spaces S(D^{2} × F^{∗}, σ̌) do not admit any Whittaker models.

\underline{Proof}: If one of these spaces admits a nontrivial Whittaker functional, then by composition with the projection also the space S(D^{2} × F^{∗}). This however is not the case, as already shown. This proves the lemma.

\underline{The Siegel parabolic}: Similarily the normalized Jaquet module with respect to the Siegel parabolic P in GSp(4, F) is isomorphic to the representation ν^{1/2} ∘ det ⊠ ν^{3/2}S(F^{∗}) of the

Levi component Gl(2, F) × Gl(1, F) of P

δ_{P}^{−1/2}π(P) = ν^{1/2} ∘ det ⊠ ν^{3/2}S(F^{∗}) .

This follows from the fact, that Q(T, X, X) = 0 for all T implies X = 0. Therefore

S(D^{2} ×F^{∗})/π(Rad(P))−1) ≅ S(F^{∗}). The isomorphism is induced by the map Φ(x_{1}, x_{2}, t) ↦ Φ(0, 0, t) ∈ S(F^{∗}). Observe that blockdiag(A, λA^{−t}) has modulus δ_{P} = |det(A)/λ|^{3} and acts

on Φ(0, 0, t) by multiplication with |det(A)|^{2} times the trivial representation on S(F^{∗}). On the other hand diag(1, 1, λ, λ) acts by the standard representation on S(F^{∗}) and has |λ|^{−3}.

By adjunction one has a natural nontrivial map from (S(D^{2} × F^{∗}), π) to the induced

representation Ind_{P}^{G}(ν^{1/2} ∘ det ⊠ (ν^{3/2} ⊗ S(F^{∗}))

(S(D^{2} × F^{∗}), π) → ν^{1/2} ∘ det ◁ ν^{3/2}S(F^{∗}) .

There are analogous maps for the representations on S(D^{2} × F^{∗}, σ̌).

\underline{Corollary 12}: The admissable representation π(σ̌) of GSp(4, F) on the space S(D^{2} × F^{∗}, σ̌), and especially all quotient representations Θ_{−}(σ̌), are cuspidal if σ̌_{1} is not isomorphic to σ̌_{2}.

\underline{Proof}: Note that the Jaquet module S(D^{2} × F^{∗}, σ̌)(P) is for all parabolic subgroups P a quotient of the Jaquet module (S(D^{2} × F^{∗}) ⊗ Vσ̌)(P), namely the one which is maximal with σ̌ action of M_{c}.

Under the assumption on σ̌ we have Φ(0, t) = 0 for all Φ(X, t) ∈ S(D^{2} × F^{∗}, σ̌). Therefore the Jaquet module with respect to the Siegel parabolic P is trivial. Note that on the other hand

S(D^{2} × F^{∗}, σ̌)(Q) ≅ {Φ ∈ S(D^{2} × F^{∗})(Q) ⊗ Vσ̌ | π̃(h)Φ = σ̌Φ , h ∈ M_{c}} .
= ν ⊗ S(D × F^{∗}, σ̌) .

Again under the assumptions of the lemma this Jaquet module vanishes. Therefore S(D^{2} × F^{∗}, σ̌) is cuspidal. The lemma is proven.

Now assume σ̌_{1} ≅ σ̌_{2} and assume dim(σ̌_{2}) > 1. The adjunction map with respect to P is then zero. Furthermore (S(D, σ̌), π(σ̌)) is isomorphic to the Jaquet-Langlands lift σ_{2} = JL(σ̌_{2}). By our assumption dim(σ̌_{2}) > 1 the representation JL(σ̌_{2}) is cuspidal. Consider the adjunction map

Ψ(σ̌) : S(D^{2} × F^{∗}, σ̌) → Ind_{Q}^{G}(1 ⊠ S(D × F^{∗}, σ̌)) = 1 × JL(σ̌_{2}) .

The representation on the right side is the direct sum

1 × JL(σ̌_{2}) = W p_{+}(JL(σ̌_{2})) ⊕ W p_{−}(JL(σ̌_{2})) .

The image of Ψ(σ̌) does not admit a Whittaker model. Therefore

\underline{Corollary 13}: Suppose σ̌_{1} ≅ σ̌_{2} and dim(σ̌_{2}) > 1. Then

Image(Ψ(σ̌)) = W p_{−}(JL(σ̌_{2}))

and the kernel of Ψ(σ̌) is an admissable cuspidal ω-representation.

The anisotropic theta lift Θ_{−} (global theory)

Consider a quaternion algebra D over the totally real number field F. Suppose S is a finite number of places including all infinite places. Suppose D ramifies at all places v ∈ S. Consider the global group M_{c} = (D^{∗} × D^{∗})/F^{∗}. Suppose there are given irreducible representations

σ̌_{v} = σ̌_{1,v} × σ̌_{2,v} , (v ∈ S)

at the places in S. Let us make the following

\underline{Assumptions}:

1) There are place v′, v′′ ∈ S such that dim(σ̌_{1,v′}) > 1 and dim(σ̌_{2,v′′}) > 1.

2) For some place v′′′ ∈ S the representations σ̌_{v′′′,i} are not isomorphic.

Suppose we are given functions

Φ_{v} ≠ 0 ∈ S_{v} ⊂ S(D_{v}^{2} × F_{v}^{∗}, σ̌_{v}) (v ∈ S) ,

where S_{v} denotes some admissable subrepresenation. Observe

S(D_{v}^{2} × F_{v}^{∗}, ω_{v}) = ⨁_{ω(σ̌_{2,v})=ω_{v}} S(D_{v}^{2} × F_{v}^{∗}, σ̌_{v}) .

Hence we view the Φ_{v} as elements of

Φ_{v} ∈ S(D_{v}^{2} × F_{v}^{∗}, ω_{v}) (v ∈ S) .

Now choose any lifts

Φ′_{v} ∈ S(D_{v}^{2} × F_{v}^{∗}) (v ∈ S) ,

under the central projection.

Then there exist points (X_{v}, t_{v}) ⊂ D_{v}^{2} × F_{v}^{∗}, where Φ′_{v}(X_{v}, t_{v}) ≠ 0. We can approximate by a global point (X, t) ∈ D^{2} × F^{∗}, such that

Φ′_{v}(X, t) ≠ 0 (v ∈ S) .

We could also assume X ≠ 0. Then there exists a function ∏_{v∉S} Φ′_{v} with sufficantly small support containing (X, t), such that Φ′ = ∏_{v} Φ′_{v} has the property

supp(Φ′) ∩ D^{2} × F^{∗} = {(X, t)} .

Therefore ϑ_{Φ′}(1, 1) = Φ′(X, t) ≠ 0. Especially

ϑ_{Φ′}(m, 1) ≠ 0

as a function of m ∈ M_{c}(A A)/M_{c}(F). By our assumption 1) there exists a cuspidal automorphic representation σ̌ = σ̌_{1} ×σ̌_{2} (i.e dim(σ̌) ≠ 1) and a function f in the representation space of σ̌ such that

L_{f}(Φ′)(g) = ∫_{M_{c}(A A )/M_{c}(F)} ϑ_{Φ′}(m, 1)\overline{f}(m)dm ≠ 0 .

The corresponding theta lift maps to the space of cuspform of G(A A) by assumption part 2). We can therefore use orthogonal projection to (any) irreducible automorphic quotient π of this theta lift. The map

pr_{π} ∘ L_{f} : S(D^{2}(A A) × A A^{∗}) → π

factors over a map

pr_{π} ∘ L_{f} : S(D^{2}(A A) × A A^{∗}, ω) → π

where ω is the central character of π (or σ̌). Restriction to S(D_{v}^{2} ×F_{v}^{∗}, σ̌_{v})⊗∏_{w≠v} Φ′_{w} gives a \underline{nontrivial} intertwining map

S_{v} → π_{v} .

As we are mainly interested in the nonarchimedian places v, we assume

\underline{Assumption 3}: v ≠ ∞ and dim(σ̌_{1,∞}) = 1 and dim(σ̌_{2,∞}) = k − 2 > 1.

Under the assumptions 1),2) and 3) we then get the following: By assumption 3) the cuspidal representation π gives cohomology (as studied in this paper). By [PSS], thm 2.4 (1) it is a weak lift of σ = σ_{1} × σ_{2}, where σ_{i} = JL(σ̌_{i}) are cuspidal (holomorphic) automorphic representations of Gl(2, F) obtained by the global Jaquet-Langlands lift. As dim(σ̌) ≠ 1 the representations σ_{i} satisfy the Ramanujan conjecture. Therefore [PSS], thm. 2.4 (1) also implies, that π is not a CAP representation. Suppose, the multiplicity conjecture after corollary 5 is true. Then π is detected by the multiplicity formula. This implies π_{v} is either π_{+}(σ_{v}) or π_{−}(σ_{v}). π_{v} does not have a Whittaker model, therefore π_{v} ≅ π_{−}(σ_{v}). Furthermore assumptions 1),2) and 3) can satisfied in order to proof the next

\underline{Corollary 14}: Assume that σ_{v} is an irreducible discrete series representation of M_{v}. Suppose that the multiplicity conjecture stated after corollary 5 is true. Then r(σ_{v}) = Θ_{+}(σ_{v}) − Θ_{−}(σ_{v}). Furthermore: Θ_{−}(σ_{v}) = π_{−}(σ_{v}) does not have a Whittaker model.

\underline{Corollary 15}: Suppose dim(σ̌_{v}) > 1. Then under the assumptions of corollary 14 above S(D_{v}^{2} × F_{v}^{∗}, σ̌_{v}) for nonarchimedian v ∈ S is isomorphic to a finite multiple of

π_{−}(σ_{v}).

\underline{Proof}: Use some additional places to satiesfy assumption 1),2) and 3) in a global context. Then under assumption of the corollary 14 every submodule S_{v} of S(D_{v}^{2} × F_{v}^{∗}, σ̌_{v}) for nonarchimedian v ∈ S has a nontrivial quotient, which must be isomorphic to π_{−}(σ_{v}).

Quotients of cuspidal admissable ω-representations are again cuspidal. In the category of smooth ω_{v} representations they are projective and therefore split as a direct summand.

The kernel of the adjunction map Ψ(σ̌_{v}) is a cuspidal, admissable ω-representation. Therefore all possible cuspidal theta images split off as direct summands. This shows, that the kernel is an isotypic multiple of π_{−}(σ_{v}) because S(D_{v}^{2} × F_{v}^{∗}, σ̌_{v}) is admissable. Therefore each summand appears with finite multiplicity. This finishes the argument unless Ψ(σ̌_{v}) is nontrivial. Then we already know, that π_{−}(σ_{v}) = W p_{−}(σ_{v}) is not cuspidal. Therefore the same argument shows, that the cuspidal representation Kern(Ψ(σ̌_{v})) would be isomorphic to d · π_{−}(σ_{v}). But this is impossible unless d = 0, because π_{−}(σ_{v}) = W p_{−}(σ_{v}) is not cuspidal. Therefore π_{−}(σ_{v}) ≅ W p_{−}(JL(σ̌)). This completes the proof.

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Four dimensional Galois Representations

Rainer Weissauer (Mannheim)

\underline{Theorem 1}: Let Π be a cuspidal irreducible automorphic representation of GSp(4, A A) such that Π_{∞} belongs to the holomorphic discrete series of weight (k_{1}, k_{2}). Put w = k_{1} + k_{2} − 3. Let S denote the set of ramified places of the representation Π. Then there exists a number field E, such that

L_{p}(p^{−s}) = L_{p}(Π_{p}, s − \frac{w}{2} ) , L_{p}(X) ∈ E[X] ,

and such that for any prime number l and any extension λ of l to E there exists a continuous Galois representation

ρ_{Π,λ} : Gal(\overline{Q} : Q) → Gl(4, E_{λ}) ,

which is unramified outside S ∪ {l} and for p ∉ S ∪ {l} the following holds

L_{p}(Π_{p}, s − \frac{w}{2} ) = det(1 − ρ_{Π,λ}(F rob_{p})p^{−s})^{−1} .

The representations ρ_{Π,λ} are mixed. They are pure of weight w if Π is not a CAP representation.

\underline{Proof}: If Π is CAP, then the existance of ρ_{Π} is reduced to the corresponding statement for Gl(2, A A) by the classification of Piateskii-Shapiro and Soudry.

If Π is not CAP, but an (endoscopic) weak lift, one again reduces to the case Gl(2, A A) as explained already.

If Π is neither CAP nor an endoscopic weak lift, then one considers the generalized eigenspace W of some near equivalence class of Π on a third etale cohomology group, as defined in the next section. We will show, that W is the multiple m · ρ_{Π} of a four dimensional representation ρ_{Π}. This representation ρ_{Π} then has the desired properties. Let us scetch the argument.

Consider the Zariski closure G of Gal(\overline{Q} : Q) in Gl(W). Then arguments of [T] give the desired result, except in two exceptional cases (case 1 and 3), where G is very small and especially rank(G) < 3. It remains to consider these two exceptional cases. The information on G and some additional information on π_{0}(G) provided by lemma 25 from the appendix imply that the representation Π is D-critical for some D. The meaning is explained below. This follows from lemma 3,4 and 7 under the hypothesis, that theorem 1 does not hold for Π. To be D-critical is a strong restriction on Π, which implies in the situation where Π satisfies the Ramanujan conjecture at almost all places, that the restriction of Π to Sp(4, A A) is a theta lift from some orthogonal group of rank 4, and hence is related to some automorphic form π on Gl(2, A A_{K}) for some quadratic extension field K of Q. See lemma 10, lemma 24 and theorem 2 together with lemma 13. A further discussion on poles of certain L-series attached to Π and π implies that D = 4 or D = 8 and furthermore that then π satisfies

σ(Π) ≅ π ⊗ χ ∘ Norm_{K}

for some character χ of A A^{∗}/Q^{∗}. Here σ denotes the nontrivial automorphism of K/Q. This is the content of theorem 4 and 5.

At this stage one should obtain a contradiction from the fact, that

σ(π_{∞}) ≅ π_{∞} ⊗ χ_{∞} ∘ Norm_{K_{∞}}

and the fact that π_{∞} and Π_{∞} are related by some explicitely given lift, where Π_{∞} is in the discrete series. It is very likely that this applies for an arbitrary discrete series representation and follows from considering only the Harish-Chandra parameters and K-types.

Suppose K is real quadratic. Suppose Π_{∞} is a representation of the discrete series, and suppose its restriction to Sp(4, R) is a Howe lift from an orthogonal group O(V) of a quadratic space V of rank 4. Then K_{∞} = R ⊕ R implies that Π_{∞} corresponds to a representation π_{∞} = π_{1} × π_{2} on the quasisplit inner form Gl(2, R) × Gl(2, R) (in the sense explained before lemma 13). We also know π_{1} ≅ π_{2} ⊗ χ_{∞}, as explained above. We can now apply [KV] if V is positive or negative definite, or [Pr] if V is split. These are the only possibilities if K_{∞} splits. Let us consider these two cases sepaeately.

First the (positive or negative) definite case. Then by [KV] the Howe theta lift belongs a holomorphic or antiholomorphic discrete series or limit of discrete series of Sp(4, R). By assumption Π_{∞} belongs to the discrete series, hence it must now belong to the holomorphic discrete series. Suppose Π_{∞} has highest weights k_{1} ≥ k_{2} ≥ 3 and assume that K is real quadratic. Then the lift gives an irreducible representation σ_{∞} = σ_{1} × σ_{2} of a product of the Hamilton quaternions H^{∗} × H^{∗} with dim(σ_{i}) > 1. σ_{∞} is related by the Jaquet-Langlands lift to a representation π_{∞} which has the same Harish-Chandra parameters as π_{∞} = π_{r_{1}} ×π_{r_{2}}, a pair of holomorphic discrete series of weights r_{1} = k_{1}+k_{2}− 2 > r_{2} = k_{1} −k_{2} +2. This excludes the possibility, that σ(π_{∞}) ≅ π_{∞} ⊗χ_{∞} ∘Norm_{K_{∞}}. Hence in this case we get a contradiction. Now suppose V is split and the orthogonal group O(V) is isomorphic to O(2, 2) in the sense of [Pr], (2.1.4). By [Pr] thm. 3.6.1 (case 3.3.1) a discrete series representation of Sp(4, R) ([Pr] (2.4.42)) corresponds to discrete series representations of Sl(2, R)×Sl(2, R) (the connected component of O(2, 2)) after restriction. Only those discrete series representations appear in the image, which are not conjugation invariant under the twisted action of the nontrivial element ϵ ∈ O(2, 2)/SO(2, 2). Note that diag(1, 1, 1, −1) is a representative in O(2, 2) for ϵ, and that conjugation by this representative is easily studied on K_{∞} types for SO(2, R)×SO(2, R) embedded into O(2, 2) as in [Pr], (2.1.4). It replaces the type (m, n) by (m, −n). This easily allows to determine the ϵ-invariant discrete series representations of SO(2, 2). In the notation of [Pr], (2.5.3) the ϵ-conjugation invariant discrete series representations of SO(2, 2) are π_{m+1,0}, π_{0,n−1}, which extend to the irreducible representations π_{m+1,0}^{p}, π_{0,n−1}^{p} (p = 0, 1) of O(2, 2). Exactly these representations are not in the image of a discrete series representation of Sp(4, R), under the Howe lift. We have to understand this in terms of the representation π on Gl(2, R)×Gl(2, R). It defines a representation of the quotient group GSO_{2,2}. Its restriction to SO_{2,2} has to be described in terms of the notations of [Pr]. We can identify SO(2, 2) and SO_{2,2} using notation of [Pr]. Then SO(2, 2) corresponds to the group SO_{2,2} of all

pairs (g_{1}, g_{2}) with det(g_{1}) = det(g_{2}) = ±1 modulo (g_{1}, g_{2}) ∼ (−g_{1}, −g_{2}) embedded in Gl(4, R)

by

(g_{1}, g_{2}) ↦ diag(g_{1}, det(g_{1})(g′_{1})^{−1}) · ( \begin{matrix}a & 0 & 0 & b \\ 0 & a & −b & 0 \\ 0 & −c & d & 0 \\ c & 0 & 0 & d \end{matrix} ) , g_{2} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

Using this isomorphism our representative for ϵ in O(2, 2) corresponds to the matrix

ε = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} )

in O_{2,2}. Obviously conjugation by ε of the image of (g_{1}, g_{2}) flips the variables g_{1}, g_{2} ↦ g_{2}, g_{1}. Thus the representation π = π_{1} × π_{2} of SO_{2,2} with π_{1} ≅ π_{2} ⊗ χ_{∞} is invariant under conjugation by ε up to isomorphism. This gives a contradiction to what was explained above.Thus theorem 1 is proved, if K can be chosen to be real quadratic.

Suppose it would be impossible to choose K to be real quadratic. Then Π_{∞} could not belong to the holomorphic discrete series. Otherwise any nonvanishing Fourier coefficient of Π with respect to a degenerate Whittaker model would correspond to a symmetric positive matrix T ∈ M_{2,2}(Q), whose discriminant character χ_{T} defines an imaginary quadratic extension field of Q (Koecher’s principle). This contradicts theorem 5 and corollary 2, which state that either χ_{T} = 1 or χ_{T} = χ_{K_{1}}χ_{K_{2}}. Both alternatives yield a contradiction and therefore proof theorem 1. It is clear that χ_{T} = 1 is impossible. It is also true that χ_{T} = χ_{K_{1}}χ_{K_{2}} is impossible. Both are by our assumption on the field K above, without restriction of generality, not real quadratic. But for imaginary quadratic fields K_{1} and K_{2}, the field K_{T} attached to χ_{K_{1}}χ_{K_{2}} would be real quadratic. This is a contradiction. This completes the sketch of the proof.

\underline{The representation W attached to Π}:

Consider a finite dimensional vector space W over k = \overline{Q_{l}}. Suppose W arises from an λadic representation of Gal(\overline{Q}/Q) by extension of scalars. Let G denote the Zariski closure of the image of the Galois group. Assume G is reductive with connected component G^{0}. Let T be a maximal split torus in G^{0}. Let R ∈ X^{∗}(T) denote the roots of T of the representation s

s : G ↪ Gl(W) .

We have a homomorphism

r : Gal(\overline{Q}/Q) → G .

We suppose

a) There exists an algebraic homomorphism

n : G → k^{∗} , n|G^{0} ≠ 1

nontrivial on G^{0}.

b) For all g ∈ G there are at most 4 eigenvalues of s(g), which come in at most two

pairs α, n(g)/α.

c) For all λ ∈ R we have λ^{2} ≠ n.

d) Π satisfies the Ramanujan conjecture for all unramified nonarchimedian places.

e) The k-dimension of W is 4m and

L_{v}(Π, s)^{m} = det(1 − s(F rob_{v})p_{v}^{−s})^{−1}

for all unramified nonarchimedian places v.

\underline{Remark}: This is the setting of [T], section 2, where W is defined from the representation of Gal(\overline{Q}/Q) on a Π-isotypic component in the third cohomology of a Siegel modular threefold. Here Π is a finite ∗-closed set of nearly equivalent irreducible cuspidal admissable representation of GSp(4, A A). Suppose S is a finite set of places containing all archimedian places and all places where the representations Π are ramified. By abuse of notation, we fix one of these representation and also denote it Π in the following. We assume

n ∘ r = ω_{Π}μ_{l}^{−3} .

We can assume, that Π is not CAP. The conditions d) and e), which we then can add, are stronger than those available in [T].

We can switch to a character twist of Π, whose central character is unitary. We do this in order to normalize the functional equation of L-series. Let us consider such a \underline{unitary} cuspidal irreducible automorphic representation Π of GSp(4, A A) with central character ω_{Π}.

\underline{Definition}: We say that Π is \underline{D-critical} if the following holds:

i) There exists a Galois extension L/Q with Galois group of degree D = [L : Q] with the following properties. Let S be a finite set of Q places containing the ramified or archimedian places of Π and L. There exists a set T of Q-places containing S of Dirichlet density zero, such that the following holds: For all v ∉ T one has

Π_{v} ∼ (ν_{v}, −ν_{v}, μ_{v}, −μ_{v}) , Satake parameters

if v does not split completely in L/Q and

Π_{v} ∼ (ν_{v}, ν_{v}, μ_{v}, μ_{v}) , Satake parameters

if v splits completely in L/Q.

ii) For v ∉ S we have ω_{Π}(p_{v}) = ν_{v}μ_{v} for the unitary central charcter ω_{Π}.

iii) The Ramanujan conjecture holds for all v ∉ S: |ν_{v}| = |μ_{v}| = 1.

Abbreviate Ad_{v} = (μ_{v}/ν_{v} + ν_{v}/μ_{v} + 1), then for v ∉ T we have |ν_{v}| = |μ_{v}| = 1 by (iii), hence

iv) The numbers Ad_{v} are real and satisfy −1 ≤ Ad_{v} ≤ 3.

This definition is motivated as follows. We will see that under the conditions a)-e) any representation Π, for which theorem 1 does not hold, is D-critical for some D ≥ 4. The field L is the Galois extension of Q attached to some quotient Δ of the finite group G/G^{0}. It is the kernel field of the surjective map Gal(\overline{Q} : Q) → Δ induced by r.

\underline{Definition}: We say that a D-critical representation Π is \underline{nondegenerate}, if

v) (ν_{v}/μ_{v})^{2D} ≠ 1 for all v ∉ T.

We say it is of \underline{(two)-abelian type}, if Δ = Gal(L/Q) is (two)-abelian.

\underline{The degree 5 L-series}: Let Π be D-critical. The spinor L-series L(Π, s) = L(Π, st, s), attached to the degree 4 standard representation of the L-group, has for v ∉ T the form

L_{v}(Π, s)^{−1} = (1 − ν_{v}p_{v}^{−s})(1 + ν_{v}p_{v}^{−s})(1 − μ_{v}p_{v}^{−s})(1 + μ_{v}p_{v}^{−s}) ,
L_{v}(Π, s)^{−1} = (1 − ν_{v}p_{v}^{−s})(1 − ν_{v}p_{v}^{−s})(1 − μ_{v}p_{v}^{−s})(1 − μ_{v}p_{v}^{−s}) ,

depending on whether v splits in L or not. Then the degree 5 standard L-series

ζ(Π, ω_{Π}, s) = ζ(Π, ⋀^{2} st, s)/L(ω_{Π}, s) attached to Π has the form

ζ^{S}(Π, ω_{Π}χ, s) = ∏_{v∉S} ζ_{v}(Π_{v}, ω_{Π_{v}}χ_{v}, s) ,

ζ_{v}(Π_{v}, ω_{Π_{v}}χ_{v}, s) = ∏_{i}(1 − c_{v,i}χ_{v}(p_{v})p_{v}^{−s})^{−1} , v ∉ T

with c_{v,i} equal to −ν_{v}^{2}, −μ_{v}^{2}, ν_{v}μ_{v}, −ν_{v}μ_{v}, −ν_{v}μ_{v}, if v does not split completely in L, respectively ν_{v}^{2}, μ_{v}^{2}, ν_{v}μ_{v}, ν_{v}μ_{v}, ν_{v}μ_{v}, if v splits completely in L. Therefore ζ_{v}(Π_{v}, 1, s) is related to the parameters (−ν_{v}/μ_{v}, −μ_{v}/ν_{v}, −1, 1, −1) if v ∉ T does not split in L and (ν_{v}/μ_{v}, μ_{v}/ν_{v}, 1, 1, 1) if v ∉ T splits completely in L.

\underline{Lemma 1}: Let Π be D-critical. Put ε_{v} = −1 if v does not split in L and ε_{v} if v splits in L. Then the local zeta factor for v ∉ T satisfies

log ζ_{v}(Π_{v}, χ_{v}, s) = (1 + ε_{v}(Ad_{v} + 1))χ(p_{v})p_{v}^{−s} + O(p_{v}^{−2s}) .

\underline{A list of possible cases}:

In [T] at the bottom of page 298 one can find a list of 11 possible cases for the group G, which can occur under the hypotheses a), b), c) and a (weaker) version of e). Under the additional assumptions d) and e) one finds in addition that the coefficient e, f in this list of [T] satisfy: e = f in case 2,4, 5 and 9 and f = 0 in case 8. This shows, that in the cases 4-11 of this list the representation s of G on W is a m-fold direct sum of a four dimensional representation s̃: G → Gl(4, k). The image of s̃ is contained in GSp(4, k) in the cases 2, 4, 5, 6, 7, 8, 9, 11 and in GO(4, k) in case 10. To proof theorem 1 it therefore remains to understand the critical cases 1 and 3 of [T], p.298ff.

From now on let us consider the cases one and three of Taylor’s list. Before we discuss these two critical cases, let us introduce some further notations. Let \overline{G} ⊂ G be the kernel of the natural map G → O, where O = {g ∈ Out(G^{0}) | s ∘ g |G^{0} ≅ s |G^{0}}. Let N be the centralizer of G^{0} in \overline{G}. Then N ∩ G^{0} = Z(G^{0}) and \overline{G} = (N × G^{0})/Z(G^{0}). We have two sequences

0 → G^{0} → \overline{G} → N/Z(G^{0}) → 0 ,
0 → N → \overline{G} → (G^{0})_{ad} → 0 .

In the two critical cases we have Z(G^{0}) = N^{0}, which is a torus over k of rank r, r = 1, 2. Therefore π_{0}(\overline{G}) ≅ N/Z(G^{0}) ≅ π_{0}(N). Let n be an integer which annihilates the group H^{2}(π_{0}(N), k^{∗}), e.g. the order of π_{0}(N). Then there exists a finite group Ñ and a central extension

0 → (μ_{n})^{r} → Ñ → π_{0}(N) → 0 ,

such that N = Ñ × N^{0}/(μ_{n})^{r} and \overline{G} = (Ñ × G^{0})/(μ_{n})^{r}. As each of the two critical cases behaves differently, we discuss them separately.

\underline{The remaining third case of [T]}:

Recall G = \overline{G} in case 3. The representation s of G restricted to G^{0} is a 8m-fold copy of the two dimensional standard representation t of G^{0} = Gl(2, k) ([T], p. 301 bottom). Therefore

s ≅ ρ ⊗ t , ρ(zg) = z^{−1}ρ(g) , z = ( \begin{matrix}z & 0 \\ 0 & z \end{matrix} ) ∈ μ_{n}

for some finite dimensional representation ρ of the finite group Ñ. It follows from property e) of the representation Π, that for every element g ∈ Ñ the matrix ρ(g) has at most two different eigenvalues ζ_{1}(g), ζ_{2}(g) with multiplicities a_{1}(g), a_{2}(g), such that a_{1}(g)+a_{2}(g) = 8m. In case, that there is only one eigenvalue we put ζ_{1}(g) = ζ_{2}(g). Consider the ratio

ζ(g) = ζ_{1}(g)/ζ_{2}(g) g ∈ Ñ .

This is welldefined up to inverse and depends only on the image of g in the quotient group π_{0}(N) = N/Z(G^{0}) ≅ Ñ/μ_{n}. The ratio can be extended to a function on G. Namely for g ∈ G with image \overline{g} in G/G^{0} ≅ N/Z(G^{0}) put ζ(g) = ζ(\overline{g}). This is well defined. ζ(g) takes values in the finite set Ω of roots of unities of order dividing twice the order of the finite group π_{0}(N), up to inverse. We can identify this with the finite set Ω of conjugacy classes in P Gl(2, k), corresponding to the classes of the matrices diag(ζ(g), 1). In this sense

ζ : G → G/G^{0} → Ω

takes values in a finite subset Ω of P Gl(2, k)/ ∼ . Any g ∈ G = (Gl(2, k) × Ñ)/μ_{n} is up to conjugation of the form

g ∼ ( \begin{matrix}u(g) & ∗ \\ 0 & v(g) \end{matrix} ) × ñ(g) , ñ(g) ∈ Ñ .

The numbers u(g), v(g) are not uniquely determined. They are unique only up to a permutation and multiplication by a common n-th root of unity. Hence their ratio u(g)/v(g) is welldefined up to inverse. This ratio only depends on the conjugacy class of the image pr(g) of g in G/N ≅ P Gl(2, k). Let Ω ⊂ P Gl(2, k) be a finite subset as above. Let G_{Ω} ⊂ G be the subset of all elements g in G, whose image in (G^{0})_{ad} = P Gl(2, k) is conjugate to an element in Ω. By [T], lemma 2 the set T = T_{Ω} of primes v with F rob_{v} in G_{Ω} has Dirichlet density zero.

\underline{Lemma 2}: Suppose Π satisfies a) - e) and that we are in case three of Taylor’s list. Then there exists a set T of Q-places of density 0, such that for v ∉ T the Satake parameters of Π_{v} are of the form

Π_{v} ∼ (ν_{v}, ν_{v}ζ(F rob_{v})^{−1}, μ_{v}, μ_{v}ζ(F rob_{v})) , ν_{v}μ_{v} = n(F rob_{v}) ,

where ζ(g) is the root of unity defined above.

\underline{Proof}: Put T = T_{Ω}. By property e) of Π the Satake parameters (ν_{v}, ν̃_{v}, μ_{v}, μ̃_{v}) of Π_{v} for v ∉

S are up to a permutation equal to v(g)ζ_{1}(g), v(g)ζ_{2}(g), u(g)ζ_{1}(g), u(g)ζ_{2}(g) where g = F rob_{v}.

The Weyl group acts transitively on the Satake parameters (ν_{v}, ν̃_{v}, μ_{v}, μ̃_{v}), which satisfy ν_{v}μ_{v} = ν̃_{v}μ̃_{v} = n(F rob_{v}). Thus we may assume that the first Satake parameter is ν_{v} =

v(F rob_{v})ζ_{1}(g). Suppose v ∉ T. Then μ_{v} = n(F rob_{v})/ν_{v} satisfies μ_{v} ≠ v(F rob_{v})ζ_{2}(F rob_{v}). Otherwise ζ_{1}(F rob_{v})ζ_{2}(F rob_{v})v(F rob_{v})^{2} = n(F rob_{v}) = ζ_{1}(F rob_{v})ζ_{2}(F rob_{v})u(F rob_{v})^{2}, because

also ν̃_{v}μ̃_{v} = n(F rob_{v}), and then u(F rob_{v})/v(F rob_{v}) ∈ Ω. Therefore in the pair ν̃_{v}, μ̃_{v} we can choose ν̃_{v} = v(F rob_{v})ζ_{2}(g). Recall that the Weyl group contains an element fixing ν_{v}, which permutes ν̃_{v} and μ̃_{v}. This proves the lemma.

\underline{Lemma 3}: Suppose Π does not satisfy the statements of the theorem 1, then Π satisfies conditions a)-e). If it belongs to case three of Taylor’s list, then the representation Π is D-critical and nondegenerate of two-abelian type for some D ≥ 4. We have Δ = Gal(L :

Q) ≅ π_{0}(G).

\underline{Proof}: Let T be as in lemma 2. Then the images of the Frobenius elements for v ∉ T generate the finite group Δ = π_{0}(N). Furthermore (μ_{v}/ν_{v})^{2D} ≠ 1 and (μ̃_{v}/ν̃_{v})^{2D} ≠ 1. This allows to reconstruct uniquely the value ζ(F rob_{v}) (up to inverse) from Π_{v} as the unique ratio of Satake parameters, whose value is in Ω. Furthermore all Satake parameters are different from each other, unless ζ(F rob_{v}) = 1 or equivalently F rob_{v} ∈ G^{0}. Therefore property e), satisfied by Π, implies that ρ is a balanced representation of Ñ (see the appendix). ρ cannot be a multiple of a 2-dimensional representation, because otherwise s and then also r is a multiple of a 4-dimensional representation and the main theorem holds for Π. Observe that the kernel Z(ρ) of the projective representation \overline{ρ} of Ñ attached to ρ is characterized by: g ∈ Z(ρ) iff ζ(g) = 1 iff g ∈ Z(G^{0}). Hence Q(ρ) = Ñ/Z(ρ) ≅ Δ. As ρ is balanced and not a multiple of a two dimensional representation, it follows from lemma 25 in the appendix, that the quotient Q = Q(ρ) of Ñ is two-abelian of order D ≥ 4. This proves that π_{0}(G) = Δ is two-abelian. Furthermore all values ζ(g) are contained in ±1, as the order of ζ(g) equals the order of the image of g ∈ Ñ in the group Q(ρ). Thus ζ(F rob_{v}) = ±1 and = +1 iff F rob_{v} ∈ G^{0}. If L is the field attached to the finite Galois group defined by Δ = G/G^{0}, then it follows from this and lemma 2, that Π is D-critical.

\underline{The remaining first case of [T]}:

In the first case the group O is ZZ/2ZZ, therefore either G = \overline{G} or [G : \overline{G}] = 2. In order to distinguish the two possibilities, call them case 1a and 1b respectively. Then G^{0} is a torus of rank one or two over the field k and \overline{G} = (Ñ × G^{0})/μ_{n}^{r}, where r = rank(G^{0}). We have N = \overline{G}. The restriction \overline{s} of the representation s of G to \overline{G} decomposes \overline{s} = ρ ⊕ ρ′, where ρ and ρ′ are representations of \overline{G}, whose restriction to G^{0} are characters χ resp. χ′ of the torus G^{0}, such that χχ′ = n. See Taylor’s list and also [T], p. 302. Recall that our assumption c) on Π implies χ^{j} ≠ (χ′)^{j} for all powers j ≥ 1. There is a natural map \overline{G} = N → N/Ñ ≅ G^{0}/G^{0}[n] ≅ G^{0}. In other words, the characters χ^{n}, χ^{′n} of the torus G^{0} extend to a pair of different characters of \overline{G}

pr = (χ^{n}, χ^{′n}) : \overline{G} → G^{0} .

For any finite subset Ω of k^{∗} the subset \overline{G}_{Ω} = pr^{−1}(Ω) has dimension strictly smaller than \overline{G}. Therefore the set T = T_{Ω} of Q-places with F rob_{v} ∈ G_{Ω} has density 0.. Elements g of \overline{G} therefore with at most two eigenvalues ζ_{1}(g), ζ_{2}(g) on ρ and at most two eigenvalues

ζ′_{1}(g), ζ′_{2}(g) on ρ′. Put ζ(g) = ζ_{1}(g)/ζ_{2}(g) and analogously define ζ′(g) = ζ_{1}(g)/ζ_{2}(g). These

are n-th roots of unity. For a given element in π_{0}(N), there always exists an element g in \overline{G}, whose image under pr is not contained in Ω, thus the eigenvalues satisfy ζ_{1}(g)^{2n} = ζ_{2}(g)^{2n}, which is different from (ζ′_{1}(g))^{2n} = (ζ′_{2}(g))^{2n} by assumption c). Therefore assumption e) implies that both ρ and ρ′ are balanced representations of the finite group Ñ. For this notion we refer to the appendix.

\underline{Lemma 4}: Let Π as in lemma 3. If it belongs to case one of Taylor’s list, then π_{0}(\overline{G}) has a two-abelian quotient Q of order ≥ 4. There exists a set T of Q-places of density zero, such that for v ∉ T and F rob_{v} ∈ \overline{G} we have

Π_{v} ∼ (ν_{v}, ν_{v}ζ(F rob_{v})^{−1}, μ_{v}, μ_{v}ζ(F rob_{v})) , ν_{v}μ_{v} = ω_{Π}(p_{v}) ,

where ζ(F rob_{v}) ∈ ±1 and ζ(F rob_{v}) = 1 if and only if the image of F rob_{v} ∈ \overline{G} in Q is trivial. If G = \overline{G}, the representation Π is D-critical with D ≥ 4. Π is nondegenerate and of two-abelian type.

\underline{Proof}: Put T = T_{Ω}, where Ω = G^{0}[2n]. We have Π_{v} ∼ (ν_{v}, ν̃_{v}, μ_{v}, μ̃_{v}) with ν_{v}μ_{v} = ν̃_{v}μ̃_{v} =

n(F rob_{v}) for v ∉ S. For v ∉ T_{Ω} suppose F rob_{v} ∈ \overline{G}. Then without restriction of generality ν_{v} = ζ_{1}(F rob_{v}) and μ_{v} = n(F rob_{v})/ν_{v}, which must be different from ζ_{2}(F rob_{v}). Otherwise

ζ_{1}(F rob_{v})ζ_{2}(F rob_{v}) = n(F rob_{v}) = ζ′_{1}(F rob_{v})ζ′_{2}(F rob_{v}), which contradicts v ∈ T_{Ω} by raising

to n-th powers: ζ_{1}(F rob_{v})^{2n} = (ζ′_{1}(F rob_{v}))^{2n}. Using the Weyl group we can achieve Π_{v} ∼

(ζ_{1}(F rob_{v}), ζ_{2}(F rob_{v}), ζ′_{1}(F rob_{v}), ζ′_{2}(F rob_{v})). Thus ζ_{1}(F rob_{v})ζ′_{1}(F rob_{v}) = ζ_{2}(F rob_{v})ζ′_{2}(F rob_{v})

or equivalently ζ(F rob_{v}) = ζ′(F rob_{v})^{−1} for all v ∉ T. This shows

Π_{v} ∼ (ν_{v}, ν_{v}ζ(F rob_{v})^{−1}, μ_{v}, μ_{v}ζ(F rob_{v}) .

For a given element g ∈ \overline{G} the values ζ(g), ζ′(g) depends only on the image of g in the finite group \overline{G}/G^{0}π̃_{0}(N). By a density argument the above equality on Frobenius elements implies

ζ(g) = ζ′(g)^{−1} , g ∈ Ñ

for all g in the finite group Ñ. Let \overline{ρ}, \overline{ρ}′ be the associated projective representations of Ñ with kernels Z(ρ) and Z(ρ′). Put Q(ρ) = Ñ/Z(ρ). Note ζ(g) = 1 is equivalent to g ∈ Z(ρ) and similar for ρ′. Therefore we have an injective homomorphism Ñ/Z(ρ) → Q(ρ) × Q(ρ′) whose composition with each projection is surjective. Thus Q(ρ) ≅ Q(ρ′). By assumption at least one (say ρ) of the two balanced representations ρ, ρ′ is not the multiple of a two dimensional representation. Recall that otherwise s and therefore r is a multiple of a four dimensional representation and the theorem holds for Π contrary to our present assumption. Then lemma 25 of the appendix implies, that the group Q(ρ) = Ñ/Z(ρ) associated to ρ is two-abelian of order D(ρ) ≥ 4. The same holds for ρ′, as we have shown Q(ρ) ≅ Q(ρ′). If G = \overline{G}, put Δ = Q(ρ) and let L be the Galois extension of Q with Galois group Δ. It follows, that Π is D(ρ)-critical. By construction it is of two-abelian type and nondegenerate. This proves lemma 4.

Now assume G ≠ \overline{G}: Then G = ⟨\overline{G}, σ⟩ whith σ^{2} ∈ \overline{G}

0 → G^{0} → G → Δ → 0 , Δ = π_{0}(G) .

σ acts nontivially on G^{0}, which in the first case is a torus of dimension one or two over k. The restriction \overline{s} of s to \overline{G} decomposes \overline{s} ≅ ρ ⊕ ρ′. The restrictions of ρ resp. ρ′ to G^{0} are given by characters χ resp. χ′ of G^{0}, where χ ≠ χ′ and χχ′ = n ≠ 1 and σ now acts nontrivially on G^{0}. See Taylor’s list [T], p.298. Put \overline{s}^{σ}(g) = \overline{s}(σgσ^{−1}), then \overline{s}^{σ} ≅ \overline{s} by assumption. This implies ρ^{σ} ≅ ρ′, ρ^{′σ} ≅ ρ, because χ^{σ} = χ′, χ^{′σ} = χ. As σ acts nontrivially on G^{0} we get \overline{s} ≅ ρ ⊕ ρ^{σ}.

We have shown that the representation ρ of \overline{G} on V_{ρ} is balanced. If ρ is an isotypic multiple of a 2-dimensional representation, then W decomposes into a multiple of 4dimensional induced representations. Let us therefore make the \underline{assumption}, that ρ is not an isotypic multiple of a 2-dimensional representation. Then by lemma 25 of the appendix the group Q = Q(ρ) attached to the balanced representation ρ of the group Ñ_{ρ} is two-abelian of order D ≥ 4.

Consider the Zariski closure \overline{G}_{ρ} of \overline{G} in Gl(V_{ρ}). We have (\overline{G}_{ρ})^{0} ≅ G_{m} and \overline{G}_{ρ} ≅ (G_{m} × Ñ_{ρ})/μ_{n}. The group Q is isomorphic to π_{0}(\overline{G}_{ρ}). Namely \overline{ρ}(g) = 1 implies ρ(g) ∈ k^{∗} · id_{V_{ρ}}, hence g ∈ (G_{m} ∩ Ñ_{ρ})(k) = μ_{n}. Because 2 · Q = 0 we can assume n = 2 and get

\overline{G}_{ρ} = (G_{m} × Ñ_{ρ})/μ_{2}
0 → μ_{2} → Ñ_{ρ} → Q → 0 .

\underline{Remark}: The central extension Ñ_{ρ} defines a class in H^{2}(Q, μ_{2}). Let a(g_{1}, g_{2}) be a representing 2-cocycle (for the inhomogenous bar resolution), normalized such that a(0, 0) = 1. Then q(g) = a(g, g) ∈ μ_{2} is a function q : Q → μ_{2} independent of the choice of cocycle.

q(g) = 1 holds iff g̃^{2} = 1 for a (any) lift g̃ ∈ Ñ_{ρ} of g ∈ Q. Note q(g_{1} + g_{2}) = q(g_{1})q(g_{2})[g_{1}, g_{2}]

is a quadratic form on Q with associated bilinear form [g_{1}, g_{2}] = g̃_{1}^{−1}g̃_{2}g̃_{1}g̃_{2}^{−1}. We have a similar situation for ρ′ = ρ^{σ}. We identify Q(ρ) and Q′ = Q(ρ′) via σ. The corresponding function q′ on Q attached to the extension class of Ñ_{ρ′} in H^{2}(Q, μ_{2}) is q′(g) = q(σgσ^{−1}).

\underline{Lemma 5}: In case 1b of Taylor’s list G^{0} is a torus of rank 2 and Δ = π_{0}(G) satisfies

0 → Q → Δ → ZZ/2ZZ → 0 , Q = Q(ρ) .

\underline{Proof}: Let us show dim(G^{0}) = 2. Otherwise G^{0} = G_{m} and σtσ^{−1} = t^{−1} for all t ∈ G^{0}. Then χ′ = χ^{σ} = χ^{−1} implies n = χχ′ = 1, contradicting the assumption n|G^{0} ≠ 1. Obviously \overline{s} : \overline{G} ↪ \overline{G}_{ρ} × \overline{G}_{ρ′}. Because dim(\overline{G}^{0}) = dim(\overline{G}_{ρ}) + dim(\overline{G}_{ρ′}), the restriction s^{0} of \overline{s} to G^{0} induces an isomorphism s^{0} : G^{0} ≅ G_{m} ×G_{m}. Under this isomorphism the action of σ on G^{0} is transfered to action on G_{m}^{2} defined by σ(t_{1}, t_{2})σ^{−1} = (t_{2}, t_{1}) for (t_{1}, t_{2}) ∈ G_{m} × G_{m}. The isomorphism of connected components induces an injection π_{0}(\overline{G}) ↪ Q(ρ) × Q(ρ′). The next step is to show that map π_{0}(\overline{G}) → Q(ρ), obtained by the composing the injection π_{0}(\overline{G}) ↪ Q(ρ) × Q(ρ′) with the first projection, is an isomorphism. It is a surjection by definition. On the other hand, an element g ∈ π_{0}(\overline{G}) is in the kernel iff ζ(g) = 1 iff ζ′(g) = ζ(g)^{−1} = 1 iff g maps to 1 in Q(ρ) × Q(ρ′) iff g = 1. This proves the claim and implies

\overline{G} ≅ (G_{m}^{2} × Ñ)/μ_{2}^{2}
0 → μ_{2} × μ_{2} → Ñ → Q(ρ) → 0 .

Recall that the group Q = Q(ρ) is two-abelian. We identify Q(ρ′) with Q in the following. This can be slightly improved to

G ≅ (G_{m}^{2} × H)/μ_{2}^{2}
0 → Ñ → H → ZZ/2ZZ → 0 .

For this it is enough to find an element σ ∈ G \ \overline{G} such that σ^{2} ∈ Ñ. In any case σ^{2} = t · g

with t ∈ G^{0}, g ∈ Ñ. By definition σtgσ^{−1} = tg, hence t^{σ}/t = g(g^{σ})^{−1} ∈ G^{0} ∩ Ñ = μ_{2}^{2}. If t = (a, b) ∈ (k^{∗})^{2}, then t^{σ}/t = (b/a, a/b), hence b = ±a. We can replace σ by σ̃ = σ(a^{−1}, 1) ∈

σ · G^{0}, such that σ̃^{2} = (1, 1) or (1, −1) is in Ñ. Now lemma 5 is proved: μ_{2}^{2} is a normal subgroup of H and Δ = H/μ_{2}^{2} ≅ π_{0}(G), which contains Q as a normal subgroup

0 → Q → Δ → ZZ/2ZZ → 0 .

\underline{Main Example}: Q has 4 elements, hence Q ≅ (ZZ/2ZZ)^{2}.

This is the most important example later. See lemma 24. Especially Δ is of order 8 and Δ is either abelian Δ = (ZZ/2ZZ)^{3} or Δ = ZZ/4ZZ × ZZ/2ZZ, or Δ is isomorphic to the dieder group of order eight generated by N, S with S^{2} = 1, N^{4} = 1, N^{S} = N^{−1}. This follows from Huppert thm. 14.10, because the quaternion group can not contain a normal subgroup isomorphic to (ZZ/2ZZ)^{2}. Furthermore Δ = (ZZ/2ZZ)^{3} iff g^{2} = 1 for all g ∈ Δ. For further recall the following facts: The dieder group D_{8} of order eight has the following normal subgroups: The center ⟨N^{2}⟩ of order two, the subgroup R = ⟨N⟩ ≅ ZZ/4ZZ, and the two subgroups P = {1, N^{2}, SN, SN^{3}} and Q = {1, N^{2}, S, SN^{2}}, which are both isomorphic to (ZZ/2ZZ)^{2}. Using the isomorphism induced by S ↦ SN, which changes the presentation, we can without restriction of generality assume that the subgroup Q ⊂ D_{8} defined above can be identified with the given group Q = Q(ρ) ⊂ Δ. There are four additional subgroups of order two generated by S, SN^{2}, SN and SN^{3}, which are not normal in Δ. The conjugacy

classes are {1}, {N^{2}}, {N, N^{3}}, {S, SN^{2}} and {SN, SN^{3}}.

\underline{Lemma 6}: Assume σ ∉ \overline{G}. Then s(σ) has either two eigenvalues t, −t or four eigenvalues t, −t, it, −it depending on whether σ^{2} ∈ G^{0} or not. Each eigenvalue occurs with equal multiplicity.

\underline{Proof}: Consider any σ ∈ G, σ ∉ \overline{G}: Then

1) (σ · (t_{1}, t_{2}))^{2} = σ^{2} · (t, t) for t = t_{1}t_{2} ∈ k^{∗}. The element (t, t) ∈ G^{0} acts on W by t · id_{W}.

2) σ^{4} acts on W by a scalar. Namely σ^{4} ∈ (G^{0})^{σ} = {(t, t) | t ∈ k^{∗}}. Especially the eigenvalues of s(σ) are of the form ±t, ±it for some t ∈ k^{∗}.

As s(σ) permutes V_{ρ} and V_{ρ′} we have T r_{W} (s(σ)) = 0. Let a, b, c, d denote the multiplicities of the four possible eigenvalues. Then a − b + ic − id = 0. This implies a = b and c = d. So if a = 0 or c = 0 we are done. If a = 0 then we can replace t by it and assume c = 0 instead. Therefore either a = b > 0, c = d = 0, or all a, b, c, d all non negative. The first case takes place iff s(σ^{2}) acts by a scalar on W. In the second case s(σ^{2}) has eigenvalues t^{2} and −t^{2} with multiplicities 2a and 2c. Note that σ^{2} ∈ G^{0} implies σ ∈ (G^{0})^{σ}. Therefore s(σ^{2}) acts by a scalar iff σ^{2} ∈ G^{0}. It remains to show a = c in the case, where σ^{2} ∉ G^{0}. Then the image of σ^{2} ∈ \overline{G} in Q = \overline{G}/G^{0} is non zero, i.e. ζ(σ^{2}) = ζ′(σ^{2})^{−1} ≠ 1, and σ^{2} has two different eigenvalues on V_{ρ} and on V_{ρ′}. The representations ρ, ρ′ are balanced. Hence each of the two different eigenvalues of σ^{2} on V_{ρ} is obtained with equal multiplicity dim(V_{ρ})/2 and similar for ρ′. We already know, that there are only two eigenvalues t^{2}, −t^{2} of σ^{2}. We get a = c and the lemma is proved.

\underline{The Satake parameters for F rob_{v} ∉ \overline{G}}: We determine the Satake parameters of Π_{v} in case

1b and give an expression of logζ_{v}(Π_{v}, χ_{v}, s) = χ_{v}(p_{v})w_{v} ·p_{v}^{−s} +O(p_{v}^{−2s}), where ζ_{v}(Π_{v}, χ_{v}, s) is

the local factor of the degree 5 standard L-series. Cum granulo salis the possible weight factors turn out to be w_{v} = 3, 1, −1, −3.

\underline{Lemma 7}: Suppose the assumptions are as in lemma 3 and suppose we are in case 1b of Taylor’s list. Then Π is D-critical where Δ is metabelian extension of a ZZ/2ZZ by a two-abelian group Q of order ≥ 4, i.e. there exists a set T of Dirichlet density 0, such that for v ∉ T the local representation Π_{v} has Satake parameters

Π_{v} ∼ (ν_{v}, ζ_{v}^{−1}ν_{v}, μ_{v}, ζ_{v}μ_{v}) , ν_{v}μ_{v} = n(F rob_{v}) ,

where ζ_{v} = ζ(\overline{F rob}_{v}) = ±1 and ζ_{v} = 1 iff F rob_{v} ∈ G^{0}. There are 5 possibilities to be distinguished

A,B: In these cases F rob_{v} ∈ \overline{G} and w_{v} = (1 + 2ζ_{v}) + q(\overline{F rob}_{v})(\frac{Ψ}{n}(F rob_{v}) + \frac{Ψ′}{n} (F rob_{v})).

C,D,E : In these cases F rob_{v} ∉ \overline{G} and ν_{v}/μ_{v} = ±i resp. −1 resp. 1. The weights are −1, 1, −3 respectively.

\underline{Proof}: Consider \underline{case A and B}. Then F rob_{v} ∈ \overline{G} = (G_{m}^{2} × Ñ)/μ_{2}^{2}. The image \overline{F rob}_{v} of F rob_{v} in Q = Ñ/μ_{2}^{2} is well defined and as in case 1a

Π_{v} ∼ (ν_{v}, ν_{v}ζ_{v}^{−1}, μ_{v}, μ_{v}ζ_{v})

with ζ_{v} = ζ(\overline{F rob}_{v}) ∈ \{±1\} (quotient of eigenvalues ζ_{1,v}, ζ_{2,v} with respect to ρ or equivalently ρ′). Let us write F rob_{v} = (t_{v}, t′_{v}) × g_{v} mod μ_{2}^{2} with (t_{v}, t′_{v}) ∈ (k^{∗})^{2} and g_{v} ∈ Ñ. Let α_{v}, ζ_{v}α_{v} resp. α′_{v}, ζ_{v}α′_{v} be the eigenvalues of ρ(g_{v}) resp. ρ′(g_{v}). They are contained in ±1, ±i. Then

ρ(F rob_{v}) ∼ t_{v} ( \begin{matrix}α_{v}E & 0 \\ 0 & α_{v}ζ_{v}E \end{matrix} )

ρ′(F rob_{v}) ∼ t′_{v} ( \begin{matrix}α′_{v}E & 0 \\ 0 & α′_{v}ζ_{v}E \end{matrix} ) ,

so without restriction of generality by changing α′_{v} to α′_{v}ζ_{v}

Π_{v} ∼ (t_{v}α_{v}, t_{v}α_{v}ζ_{v}^{−1}, t′_{v}α′_{v}, t′_{v}α′_{v}ζ_{v}) .

\underline{Remark}: Obviously α_{v}^{2} = q(\overline{F} rob_{v}) and α′_{v}^{2} = q′(\overline{F} rob_{v}), where

q, q′ : Q → μ_{2}

are the quadratic forms on Q defined earlier.

Note that n(F rob_{v}) = ν_{v}μ_{v} = (t_{v}α_{v})(t′_{v}α′_{v}). Therefore

t_{v}α_{v}/t′_{v}α′_{v} = (t_{v}α_{v})^{2}/n(F rob_{v}) = t_{v}^{2}q(\overline{F} rob_{v})/n(F rob_{v})

and t_{v}α_{v}/t′_{v}α′_{v} = n(F rob_{v})(t′_{v}α′_{v})^{−2} = t′_{v}^{−2}q′(\overline{F} rob_{v})n(F rob_{v}). This implies (q′/q)(\overline{F} rob_{v}) =

(t_{v})^{2}(t′_{v})^{2}/n^{2}(F rob_{v}). The assignments (t, t′) × g mod μ_{2}^{2} ↦ t^{2} resp. (t′)^{2} are welldefined characters

Ψ, Φ : \overline{G} → k^{∗} .

Define Ψ′ : \overline{G} → k^{∗} by Ψ′ = n^{2}/Ψ. Note that q′(g)q(g) is a character of Q or \overline{G}, such that Ψ′ = q′qΦ. Let L be the fixed field of the inverse image of G^{0} in Gal(\overline{Q} : Q). Similar let L_{Q} be the fixed field of the inverse image of \overline{G} in Gal(\overline{Q} : Q).

\underline{Lemma 8}: Ψ/n and Ψ′/n are algebraic grossencharacters of the quadratic extension field L_{Q} of Q, which are not of finite order.

\underline{Proof}: The group \overline{G} and the isomorphism \overline{G} ≅ (G_{m} × Ñ)/μ_{2}^{2} are defined over some finite extension field E_{λ} of Q_{l} in k, such that r : Gal(\overline{Q} : L_{Q}) → \overline{G}(k) maps already into the E_{λ}-rational points of this group. Therefore Ψ ⊕ Ψ′ defines an abelian semisimple λ-adic Galois representation. In order to proof the first part of the claim it is enough to show, that Ψ ⊕ Ψ′ defines an E-rational λ-adic representation of Gal(\overline{Q} : L_{Q}) for some number field E. See [He], p.113. Choose a number field E = E_{Π} as in [T], p.297 such that the characteristic polynomial L_{v}(X) of r(F rob_{v}) is a polynomial in E[X]. This follows from

assumption e). If v splits in L, then Ψ(F rob_{v}) = χ^{2}(F rob_{v}) and Ψ′(F rob_{v}) = (χ′)^{2}(F rob_{v}) and L_{v}(X) = (1−χ(F rob_{v})X)^{2}(1−χ′(F rob_{v})X)^{2}. Hence χ(F rob_{v})+χ′(F rob_{v}), χχ′(F rob_{v}) ∈ E and also χ^{2}(F rob_{v}) + (χ′)^{2}(F rob_{v}), (χ)^{2}(χ′)^{2}(F rob_{v}) ∈ E. If v does not split in L, then

L_{v}(X) = (1 − t_{v}α_{v}X)(1 − t_{v}α_{v}ζ_{v}^{−1}X)(1 − t′_{v}α′_{v}X)(1 − t′_{v}α′_{v}ζ_{v}X)
= 1 − q(\overline{F rob_{v}})(Ψ(F rob_{v}) + Ψ′(F rob_{v}))X^{2} + ΨΨ′(F rob_{v})X^{4} ,

with q(\overline{F rob_{v}}) = ±1. Again (Ψ(F rob_{v}) + Ψ′(F rob_{v})) and ΨΨ′(F rob_{v}) are in E. Assumption

c) on the characters χ and χ′ of the torus G^{0} imply that the Grossencharacters Ψ/n and Ψ′/n are not Dirichlet characters. This proves lemma 8.

The local term logζ_{v}(Π_{v}, 1, s) for v ∉ T contributes to the global summation the term (Ad_{v} + 2)p_{v}^{−s} + O(p_{v}^{−2s}) (\underline{case A}) resp. −Ad_{v}p_{v}^{−s} + O(p_{v}^{−2s}) (\underline{case B}), depending on whether v splits in L or not. We get

Ad_{v} = 1 + \frac{t_{v}α_{v}}{t′_{v}α′_{v}} + \frac{t′_{v}α′_{v}}{t_{v}α_{v}}

and using the formulas above

Ad_{v} = 1 + q(\overline{F rob}_{v})(\frac{Ψ}{n}(F rob_{v}) + \frac{Ψ′}{n} (F rob_{v})) .

Ψ/n and Ψ′/n are not Dirichlet characters and q is constant on the cosets Gal(\overline{Q} : L_{Q})/Gal(\overline{Q} : L) ≅ Q. Therefore Ad_{v} behaves in certain averages as if Ad_{v} = 1. Hence in the average we have weights equal to 3 resp. 1 in case A resp. B.

\underline{The cases C,D and E}: Now suppose F rob_{v} ∉ \overline{G}. First we have the \underline{case C}, where F rob_{v}^{2} ∉ G^{0}. Then

Π_{v} ∼ (t_{v}, −t_{v}, it_{v}, −it_{v})

up to replacement of i by −i. Thus ν_{v}/μ_{v} = ±i. Furthermore ζ_{v}(Π_{v}, 1, s) is attached to

(1, −1, −1/i, −i, −1) and therefore log ζ_{v}(Π_{v}, 1, s) = −p_{v}^{−s} + O(p_{v}^{−2s}). The next \underline{case D} is,

where F rob_{v}^{2} ∈ G^{0} and

Π_{v} ∼ (t_{v}, −t_{v}, −t_{v}, t_{v}) .

Then ν_{v}/μ_{v} = −1 and ζ_{v}(Π_{v}, 1, s) is attached to (1, 1, −1, −1, 1), hence log ζ_{v}(Π_{v}, 1, s) =

p_{v}^{−s} + O(p_{v}^{−2s}). Finally there is the \underline{case E}, where F rob_{v}^{2} ∈ G^{0} and

Π_{v} ∼ (t_{v}, −t_{v}, t_{v}, −t_{v}) .

Then ν_{v}/μ_{v} = 1 and ζ_{v}(Π_{v}, 1, s) is attached to (1, −1, −1, −1, −1), hence log ζ_{v}(Π_{v}, 1, s) = −3p_{v}^{−s} + O(p_{v}^{−2s}).

\underline{The theta lift}:

From now on we implicitly assume, that we are in the situation of lemma 3, 4 or 7. Hence Π is D-critical of order ≥ 4. The field extension L/Q attached to Π is abelian (case 3 or 1a) or metabelian (case 1b). This will imply that the restriction of Π to Sp(4, A A) is a theta lift. This is immediate if Δ is abelian, the metabelian case will be stated and the proof in this case will be given later. We prefer to first explain the implication of this fact.

\underline{Lemma 9}: Suppose Π is D-critical and χ is a unitary character of A A^{∗}/Q^{∗}. Then for s → 1^{+} the asymptotic behaviour is

log ζ^{S}(Π, χ, s) ∼ ∑_{v not L−split} −Ad_{v} · χ(p_{v})p_{v}^{−s} + ∑_{v L−split}(Ad_{v} + 2) · χ(p_{v})p_{v}^{−s} .

It follows, that log ζ^{S}(Π, χ, s) ∼ log L^{S}(χ, s) + ∑_{v} ε_{v}(Ad_{v} + 1) · χ(p_{v})p_{v}^{−s}, where ε_{v} = −1 for

v not split in L and ε_{v} = 1 for v split in L. Furthermore if Π is D-critical of abelian type

log( ∏_{χ∈Δ̂} ζ(Π, χ, s)) ∼ D ∑_{v L−split}(Ad_{v} + 2)p_{v}^{−s} .

The product is extended over all D characters χ of the Galois group Δ of L/Q. Here ∼ means equality up to some function of asymptotic behaviour o(logζ(s)).

\underline{Proof}: This follows from lemma 1 and property iii) of Π.

Consider ord_{s=1}ζ^{S}(Π, χ, s). Assumption (iv) on Π implies, that the summands on the right in the product of lemma 9 can be estimated below by p_{v}^{−s} because

Ad_{v} + 2 ≥ 1 .

The set of primes, which are completely split in L were denoted L-split. They have Dirichlet density equal to 1/D. We get for s → 1^{+} that log(∏_{χ} ζ(Π, χ, s)) ≻ logζ(s), where ≻ means ≥ up to some function growing like o(logζ(s)).

\underline{Lemma 10}: Let Π be as in lemma 3 and G and Δ be as in case 1 or 3 of Taylor’s list. Then for at least one character χ_{0} in Δ̂ the function ζ(Π, χ_{0}, s) has a pole at s = 1.

\underline{Remark}: From [S], thm. 2.4 (see also [KRS] introduction) a character χ_{0} such that ζ^{S}(Π, χ_{0}, s) has a pole at s = 1 is necessarily quadratic χ_{0}^{2} = 1.

\underline{Proof of lemma 10}: If we are in case 3 or 1a this follows from lemma 9. Namely Π is D-critical, nondegenerate and of two-abelian type with D ≥ 4. This follows from lemma 3 and 4. If we are in case 1b, the argument is similar but slightly more involved and will therefore be postponed to lemma 24 ff.

Lemma 10 has the following important consequence. Let T ∈ Symm^{2}(Q), det(T) ≠ 0 belong to a nondegenerate non vanishing Fourier coefficient of Π as in [KRS], p.531. Such T always exist and T defines a binary quadratic space V_{T} over Q with character χ_{T} = (., Δ(V_{T})). Here Δ(V) = (−1)^{dim(V)}det(V) denotes the discriminant of a quadratic space V_{T}.

\underline{Theorem 2}([KRS], thm. 7.1): If ζ^{S}(Π, χ_{0}, s) has a pole at s = 1 (for sufficantly large finite set S and some quadratic character χ_{0}), then it is a simple pole and the restriction of Π to Sp(4, A A) is a theta lift coming from some automorphic representation on the orthogonal group O(V, A A), where V is a 4 dimensional quadratic space V_{T} ⊕ V_{T′}. Furthermore V_{T′} is a binary quadratic space with quadratic character χ_{0}.

Let denote K the rank two commutative algebra over Q attached to the quadratic character χ_{K} = χ_{T} χ_{0} by class field theory. Note, that χ_{K} is the quadratic character attached to the discriminant of V = V_{T} ⊕ V_{T′}.

\underline{The orthogonal similitude group GSO(V)}:

Let us consider orthogonal groups over a local or global field k of characteristic zero. Let V be a quadratic space over the field k. Let GSO(V) denote the connected component of the group of similitudes of V. Then

0 → SO(V)(k) → GSO(V)(k) → M(k) → 0 , (k^{∗})^{2} ⊂ M(k) ⊂ k^{∗} .

There are two cases to be distinguished. The \underline{first}, where χ_{K} = χ_{T} χ_{0} = 1 is trivial, and the \underline{second} case, where the character χ_{K} = χ_{T} χ_{0} is attached to some proper quadratic field extension K/k. We can combine the two cases, if we set K = k^{2} in the first case.

\underline{Lemma 11}: GSO(V) ≅ Res_{K/k}(D_{K}^{∗})/A, where D_{K} is a central simple algebra of rank 4 over K, and A ⊂ Res_{K/k}(K^{∗}) is the norm 1 subgroup of the center. (If K = k^{2} this is understood as GSO(V) ≅ (D^{∗})^{2}/{(t, t^{−1}) | t ∈ k^{∗}} for some central simple algebra D of rank 4 over k).

\underline{Proof}: In the first case see [HST]. In the second case a base change from k to K reduces to the first case GSO(V) ×_{k} K ≅ (D_{K}^{∗})^{2}/K^{∗}. Thus the given k-group GSO(V) is a K/k form of G = Res_{K/k}(D_{K}^{∗})/A. I claim, that it is an inner K/k-form. Over \overline{k} the groups are

isomorphic to M = Gl(2, k)^{2}/k^{∗}. Both forms map to χ_{K} ∈ Hom(G_{k}, ±1) ≅ H^{1}(k, Out(M_{a}d).

The automorphism group Aut(M) embeds into Aut(Z_{M})×Aut(M_{ad}), because M = Z_{M} ·M_{ad}.

Therefore Out(M) ↪ Aut(Z_{M}) × Out(M_{ad}). But Aut(Z_{M}) ≅ ZZ/2ZZ and Out(M_{ad}) ≅ ZZ/2ZZ.

The twisting cocycle with values in Aut(M) induces a twisting cocycle with values in Aut(Z_{M}) by restriction. A corresponding k-form G has a cocycle with trivial image

under H^{1}(k, Aut(M)) → H^{1}(k, Out(M)) → H^{1}(k, Aut(Z_{M})) if and only if Z_{G} is a split torus

of rank one. As the forms GSO(V) and Res_{K/k}(D_{K}^{∗})/A have split center, the claim follows. Put H = P D_{K}^{∗} with H(K) = D_{K}^{∗}/K^{∗}. The inner \overline{k}/k-forms of G are given by H^{1}(k, Int(G)) = H^{1}(k, Res_{K/k}(H). This equals H^{1}(K, H) with by Shapiro’s lemma. By Hilbert theorem 90 H^{1}(K, H) injects into H^{2}(K, G_{m})) ≅ Br(K). As H(\overline{K}) ≅ D_{K}^{1}(\overline{K})/\{±1\} the image factors over H^{2}(K, {±1}), hence lands in the two torsion subgroup Br(K)_{(2)} of the Brauer group. The groups Res_{K/k}(D_{K}^{∗})/A with varying algebra D_{K} over the field K (which is fixed) are non isomorphic inner k forms. The D_{K} parametrise Br(K)_{(2)}, hence the k-groups Res_{K/k}(D_{K}^{∗})/A exhaust the inner k-forms of G. Especially GSO(V) is one of them. This proves lemma 4.

\underline{Lemma 12}: The k valued points of G = Res_{K/k}(D_{K}^{∗})/A are G(k) = (D_{K}^{∗}(K) · Z_{G}(k))/A.

\underline{Proof}: As in [HST], section 1 thegroup Z_{G}(k) = k^{∗} has the same image in H^{1}(k, A) under

the edge morphism G(k) → H^{1}(k, A) as G(k). Therefore G(k) = Z_{G}(k)·im(Res_{K/k}(D_{K}^{∗}))(k).

\underline{Remark}: The pull back of an irreducible admissable representation π of GSO(V)(k) to D_{K}(K)^{∗} has central character ω_{π} ∘Norm_{K/k} of K^{∗}, where ω_{π} denotes the central character of π.

\underline{The inverse lift to GSp(4)}:

Suppose we are given an irreducible cuspidal automorphic form Π of GSp(4, A A). Let Π^{S} be the restricted tensor product over its unramified constituents v ∉ S.

Let Π̃ be the restriction of Π to Sp(4, A A). Similarily define Π̃^{S}. If Π_{i} are both irreducible cuspidal, then the following statements are equivalent: Π̃_{1}^{S} ≅ Π̃_{2}^{S} or ζ^{S}(Π_{1}, τ, s) = ζ^{S}(Π_{2}, τ, s) or Π_{1}^{S} ≅ Π_{2}^{S} ⊗ χ^{S} for some character χ^{S} of (A A^{S})^{∗}.

If Π̃ is a lift of a form π̃ of O(V)(A A), then this correspondence can be enlarged such that Π restricted to Sp(4, A A) · A A^{∗} ⊂ GSp(4, A A) is a lift of π, a representation of O(V)(A A) · A A^{∗} ⊂ GO(V)(A A). This is to be achieved with equality of central characters ω_{Π} = ω_{π}. So we consider the passage

1.) from GSp(4)(A A) to Sp(4, A A)A A^{∗} (the restriction need not to be irreducible)

2.) to GO(V)(A A)

3.) to GSO(V)(A A) = SO(V)(A A) · A A^{∗} using lemma 4 (the restriction need not be irreducible. However in the spherical case, there is a unique spherical constituent in the restriction [HST], p.381)

4.) to D_{K}(A A_{K})^{∗} (pull back) under D_{K}(A A_{K})^{∗} → GSO(V)(A A)

5.) to Gl(2, A A_{K}) (Jaquet-Langlands lift).

This is to be understood in the following way. If some restriction decomposes, we take any constituent. In the spherical case, we take spherical constituents. We denote the representation obtained on Gl(2, A A_{K}) again π by abuse of notation. Furthermore we write π = π(Π). More precisely: The representation Π gives rise to a pair

(π, ω) , ω = ω_{Π}

where π is an irreducible automorphic representation of Gl(2, A A_{K}) with central character

ω_{π} = ω ∘ Norm_{K}.

Conversely to π the theta lift assigns via the Howe correspondance [V] an automorphic representation Π′. Π′ and Π have the same restriction to Sp(4, A A). Thus Π′ is cuspidal. Furthermore

(Π′)^{S} ≅ Π^{S} ⊗ χ^{S} , (χ^{S})^{2} = 1
ω_{Π′} = ω_{Π} .

Note that we do not care, whether the unramified quadratic character χ^{S} is automorphic.

\underline{Lemma 13}: If the representation π of Res_{K/k}Gl(2, K)(A A) is assigned to Π, then there exists a cuspidal automorphic representation Π′ of GSp(4, A A) as above, such that for all spherical constituents Π′_{v} the local constituent π_{v} is spherical and the standard L-series coincide

L_{v}(Π′_{v}, s) = L_{v}(π_{v}, s) .

\underline{Proof}: Similar to [HST], lemma 10 and 11 or [V]. See also [HST], prop. 3. Observe that our conventions on theta lift are different then those used in [HST], p.387. In our sense Π and π are associated, iff Π and the contragredient π^{∗} are associated in the sense of [HST]. Our conventions imply, that Π and π have the same central character.

\underline{Remark}: The restriction of a spherical local irreducible representation Π_{v} from GSp(4, k_{v}) to Sp(k_{v})·k_{v}^{∗} determines Π_{v} only up to a twist Π′_{v} with the unramified quadratic character χ_{v}. Similarily if K_{v} ≠ k_{v}, then π_{v} resp. L_{v}(π_{v}, s) does not determine Π′_{v}. Namely there is the freedom of extracting roots ([HST], lemma 10). The problem arises, if we replace \sqrt{α} by its negative, but keep \sqrt{β} in loc. cit. This ambiguity is related to the fact, that the map Gl(2, K_{v}) → GSO(k_{v}) is not surjective and has index two in this case. However the pair (π_{v}, ω_{v}) determines Π′_{v} in the spherical case.

Suppose K to be an algebra of rank two over Q with involution σ. Suppose we are given (π, ω), where π is automorphic on Gl(2, A A_{K}) and ω is automorphic on A A^{∗}, such that ω_{π} = ω ∘ Norm_{K}. Let S denote a large set of places of Q, which contains all archimedian places, all places were π, ω and K/Q are ramified. Suppose Π′ = Π′(π, ω) is a theta lift, an irreducible automorphic representation of GSp(4, A A). Let the Satake parameters of π_{v}, v ∉ S be given as follows:

\underline{v is K-split}: π_{v} = π_{1,v} × π_{2,v} is given by Satake parameters α_{v}, β_{v} and α′_{v}, β′_{v} with

α_{v}β_{v} = α′_{v}β′_{v} = ω(p_{v}).
\underline{v is K-inert}: π_{v} is given by the Satake parameters α_{v}, β_{v} with α_{v}β_{v} = ω(p_{v}^{2}).

Then the thetalift Π′ is given at v ∉ S by Satake parameters

\underline{v is K-split}: Π′_{v} has Satake parameters α_{v}, α′_{v}, β_{v}, β′_{v} with α_{v}β_{v} = α′_{v}β′_{v} = ω(p_{v}).

\underline{v is K-inert}: Π′_{v} has Satake parameters α_{v}^{1/2}, α_{v}^{1/2}χ_{K}(p_{v}), β_{v}^{1/2}, β_{v}^{1/2}χ_{K}(p_{v}) with

α_{v}^{1/2}β_{v}^{1/2} = ω(p_{v}).

See [HST] and [V].

\underline{Lemma 14}: Suppose K, π, ω, Π′ are as above. Furthermore assume π satisfies the Ramanujan conjecture |α_{v}| = |β_{v}| = 1 for all v ∉ S (and analogously for α′_{v}, β′_{v} if defined). Let χ be a unitary idele class character of A A^{∗}. Then

ord_{s=1}(ζ^{S}(Π′, χ, s)ζ^{S}(Π′, χχ_{K}, s))

equals

ord_{s=1}(L^{S}(χ, s)L^{S}(χχ_{K}, s)L_{K}^{S}(σ(π) × π^{∗} ⊗ (χ ∘ Norm_{K}), s)) .

Hence ζ^{S}(Π′, χ, s)ζ^{S}(Π′, χχ_{K}, s) has never zeros at s = 1 and has poles at s = 1 if χ = 1, χ_{K}.

\underline{Proof}: The local L-factor of ζ^{S}(Π′, χ, s) for a given v has five Euler factors attached to the set of numbers

\{χ_{v}, χ_{v} \frac{α′_{v}}{α_{v}} , χ_{v} \frac{α′_{v}}{β_{v}} , χ_{v} \frac{β′_{v}}{α_{v}} , χ_{v} \frac{β′_{v}}{β_{v}} \} ,

if v is split in K/Q. Otherwise it is attached to the numbers

\{χ_{v} , χ_{K,v}χ_{v} , χ_{K,v}χ_{v} \frac{α_{v}^{1/2}}{β_{v}^{1/2}} , χ_{K,v}χ_{v} \frac{β_{v}^{1/2}}{α_{v}^{1/2}} , χ_{K,v}χ_{v}\} .

By the Ramanujan conjecture, the K-inert therefore do not contribute to the order of

ζ^{S}(Π′, χ, s)ζ^{S}(Π′, χ_{K}χ, s) at s = 1. The same is true for L_{K}^{S}(σ(π) ⊗ (π^{∗} ⊗ (χ ∘ Norm_{K})), s)).

So it is enough to compare K split places v. Note that σ(π^{S}) is given by replacing α_{v}, β_{v} by α′_{v}, β′_{v} and vice versa. Note that α_{v}β_{v} = α′_{v}β′_{v} implies

\{χ_{v}, χ_{v} \frac{α′_{v}}{α_{v}} , χ_{v} \frac{α′_{v}}{β_{v}} , χ_{v} \frac{β′_{v}}{α_{v}} , χ_{v} \frac{β′_{v}}{β_{v}} \} = \{χ_{v}, χ_{v} \frac{β_{v}}{β′_{v}} , χ_{v} \frac{α_{v}}{β′_{v}} , χ_{v} \frac{β_{v}}{α′_{v}} , χ_{v} \frac{α_{v}}{α′_{v}} \} .

This finishes the proof.

\underline{Proposition 1}: For K, π, S in the last lemma

ord_{s=1}(ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)) = ord_{s=1}(L_{K}^{S}(χ ∘ Norm_{K}, s)L_{K}^{S}(σ(π) × π^{∗} ⊗ (χ ∘ Norm_{K}), s)) .

Hence ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s) has no zero at s = 1. Furthermore it has poles at s = 1 if χ = 1, χ_{K}. If K is a field and π is cuspidal, then χ ≠ 1, χ_{K} is a pole iff σ(π) ≅ π^{∗} ⊗ (χ ∘ Norm_{K}).

\underline{Proof}: This follows from lemma 6, because ζ^{S}(Π, χ, s) = ζ^{S}(Π′, χ, s) and because Π and therefore π satisfy the Ramanujan conjecture for all v ∉ S. The right side is never zero at s = 1. Finally see [AC], p.200.

\underline{The representation π of Gl(2, A A_{K})}:

Let K be a rank two algebra over Q, hence either a quadratic field extension or the sum Q ⊕ Q.

\underline{Assumption}: To write down π assume for simplicity χ = 1.

Otherwise one has to replace the given formulas for π below by their twist under χ. The form π constructed from Π respectively honestly speaking from Π′ = Π ⊗ χ, has the following form. Let v ∉ S run over the primes of Q. Then there are the following cases, where we list the L-parameters of the spherical representations π_{v}. For all v ∉ S the representation π satisfies the Ramanujan conjecture. Furthermore

\underline{K-split cases}: For v ∉ T under the simplifying assumption Π = Π′

a) ( \begin{matrix}ν_{v} & 0 \\ 0 & μ_{v} \end{matrix} ) × ( \begin{matrix}ν_{v} & 0 \\ 0 & μ_{v} \end{matrix} ), when v splits completely in K · L.
b) ( \begin{matrix}ν_{v} & 0 \\ 0 & μ_{v} \end{matrix} ) × ( \begin{matrix}−ν_{v} & 0 \\ 0 & −μ_{v} \end{matrix} ), when v splits in K, but not in L.

\underline{K-inert cases}: These cases are of less importance, because the norm of the prime w in K lying over v is (p_{v})^{2}. Because of property iii) of Π their contribution to the asymptotic behaviour of L-series at s → 1^{+} is neglectible.

c) ( \begin{matrix}ν_{v}^{2} & 0 \\ 0 & μ_{v}^{2} \end{matrix} ), when v does not split in K.

\underline{Proposition 2}: Assume we are in case 3 or 1a, so Π is nondegenerate. Let σ be the involution of the algebra K/Q. 1)Then σ(π) is not isomorphic to π. 2)Furthermore if D ≥ 8 or D = 4 and LK ≠ L, then σ(π) ≅ π̸ ⊗ χ for all characters χ of A_{K}^{∗}/K^{∗}. 3) If D = 4 and LK = L and K is a field, then σ(π) ≅ π ⊗ χ ∘ Norm_{K} for χ ≠ 1, χ_{K} in Ĝ.

\underline{Proof}: We can use a strong form of multiplicity one for π or π^{S} ⊗ χ^{S}, because both satisfy the Ramanujan conjecture at almost all places. Therefore it is allowed to discard any set T of places of Dirichlet density zero in K. We can replace Π by Π′, because the twist does not affect the following arguments.

Statement 1) is clear because the set of places v which split in K but not in L is nonempty by D ≥ 4. For such places we have ν_{v} ≠ −ν_{v}, −μ_{v}, because of the nondegenerateness property v) of Π. This implies σ(π) ≅ π̸.

Statement 2) is shown similarily. Consider the same Q-places v as in the proof of statement 1). Under the assumptions of 2) the set of K-places above these Q-places has K-density ≥ 3/4. Furthermore for the extensions w, w′ of such a Q-place v we get

from σ(π) ≅ π ⊗ χ either (−ν_{v}, −μ_{v}) = (ν_{v}χ_{w}, μ_{v}χ_{w}) or (−μ_{v}, −ν_{v}) = (ν_{v}χ_{w}, μ_{v}χ_{w}). This

implies either χ_{w} = −1 or χ_{w} = −μ_{v}/ν_{v} = −ν_{v}/μ_{v}, in any case χ has to be a quadratic character χ_{w}^{2} = 1. But property v) allows the second case only for v ∈ T. Thus for a set of K-density > 1/2 the quadratic character χ has the value −1. This is impossible. Statement 2) follows.

Statement 3) is proved as follows: For χ as in the statement, σ(π)_{w} ≅ π_{w} ⊗ χ_{v} ∘ Norm_{K} for all w lying over Q-places v ∉ T, which split in K/Q. This is a set of K-places of K-density 1. As π satisfies the Ramanujan condition, this implies σ(π) ≅ π ⊗ χ ∘ Norm_{K}.

\underline{Lemma 15}: Assume, that we are in case 3 or 1a, hence Π is nondegenerate. Then π^{S} satisfies π^{S} ≅ π̸^{S} ⊗ χ^{S} for any character χ ≠ 1 in A A_{K}^{∗}/K^{∗}.

\underline{Proof}: Consider the set of places v ∉ T, which split in K/Q. The K-extensions w of such places have K-density 1. Any character χ satisfying π ≅ π ⊗ χ is necessarily a quadratic character, namely either χ_{w} = 1 or χ_{w} = μ_{v}/ν_{v} = ν_{v}/μ_{v}. Again property v) implies χ = 1.

\underline{The adjoint L-series of π}:

Remember that the adjoint L-series ζ^{S}(Ad(π), χ, s) does not change under a character twist of π. They are uniquely determined by Π. Namely if we replace Π by Π′, we also replace π^{S} by π^{S} ⊗ χ^{S} ∘ Norm_{K}.

\underline{Lemma 16}: Let π be the automorphic form on Gl(2, A A_{K}) attched to Π as in theorem 2.. For v ∉ T the local L-factor of ζ^{S}(Ad(π), 1, s) is

v splits in K: For both extensions w of the place v we have the same local factor

ζ_{w}(Ad(π_{w}), 1, s)^{−1} = (1 − \frac{ν_{v}}{μ_{v}} p_{v}^{−s})(1 − p_{v}^{−s})(1 − \frac{μ_{v}}{ν_{v}} p_{v}^{−s})

v inert in K:

ζ_{v}(Ad(π_{v}), 1, s)^{−1} = (1 − \frac{ν_{v}^{2}}{μ_{v}^{2}} p_{v}^{−2s})(1 − p_{v}^{−2s})(1 − \frac{μ_{v}^{2}}{ν_{v}^{2}} p_{v}^{−2s}) .

\underline{Lemma 17}: Let K and π be as above. Let χ be any idele class character of A A_{K}^{∗}, (the case K = Q ⊕ Q is included!). Then at s → 1^{+} we have

logζ^{S}(Ad(π), χ, s) ∼ ∑_{v K−split} Ad_{v} · (χ_{v} + χ′_{v})p_{v}^{−s} .

The sum runs over all places v ∉ S of Q. Here χ_{v}, χ′_{v} denote the two values of χ_{w}(π_{w}) for the two places w of K over v.

If we apply this with χ ∘ Norm_{K} instead of χ then χ_{v} + χ′_{v} = 2χ_{v}(p_{v}).

\underline{Proposition 3}: Assume K to be a field. We have for an irreducible automorphic representation π of Gl(2, A A_{K}) and idele class character χ of A A_{K}/K^{∗}

ζ^{S}(Ad(π), χ, s)L_{K}^{S}(χ, s) = L^{S}(π^{∗} × π ⊗ χ, s) .

If π is cuspidal, then the right side is non zero at s = 1 at has at most a simple pole in case χ = 1. The same is true for L^{S}(χ, s), hence ζ^{S}(Ad(π), χ, s) has no zero at s = 1 and not pole at s = 1 except for character χ ≠ 1, such that π ⊗ χ ≅ π (CM-case). If π is not cuspidal, then the right side is still nonvanishing at s = 1. The case K = Q ⊕ Q is similar.

\underline{Proof}: See [AC], p.200.

\underline{Theorem 3}: If π is a irreducible automorphic representation of Gl(2, A A_{K}). Let χ be an idele character of A A_{K}^{∗}. Then the partial L-function ζ^{S}(Ad(π), χ, s) has no zero at s = 1. Suppose K is a field, then for any finite set X of such characters

ord_{s=1} ∏_{χ∈X} ζ^{S}(Ad(π), χ, s) ≤ 3 .

If π is \underline{cuspidal} and K is a field, then ζ^{S}(Ad(π), χ, s) has poles at s = 1 only if χ ≠ 1 with

π ≅ π ⊗ χ.

\underline{Proof}: This is a well known fact. See [LL] and [GJ].

\underline{Lemmas concerning poles}:

\underline{Step 1}: We apply lemma 9 and get for χ, χχ_{K} with χ ∈ Δ̂

log(ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)) ∼

∑_{v K−split, not L−split} −2 · Ad_{v}χ_{v}(p_{v})p_{v}^{−s} + ∑_{v K−split, L−split} 2 · (Ad_{v} + 2)p_{v}^{−s} .

\underline{Step 2}: On the other hand by lemma 17 and χ ∈ Δ̂

log ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) ∼ ∑_{v K−split} 2 · Ad_{v}χ_{v}(p_{v})p_{v}^{−s} .

The sum runs in both cases over places of Q. Combining Step 1 and 2 gives

\underline{Lemma 18}: For D-critical Π and χ ∈ Δ̂ the following holds

log(ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)ζ^{S}(Ad(π), χ ∘ Norm_{K}, s)) ∼ ∑_{v K−split, L−split} 4 · (Ad_{v} + 1)p_{v}^{−s} .

Especially ord_{s=1}ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)ζ^{S}(Ad(π), χ∘Norm_{K}, s) = n with an integer 0 ≤ n ≤

16/[KL : Q], which is independent from χ ∈ Δ̂.

Here we used the following \underline{notation}: KL denotes the field, generated by K and L if K is a field, and denotes the field L, if K is split.

\underline{Proof}: It is obvious, that the right side is independent from χ. The right side is a sum over set of Q-primes of Dirichlet density [KL : Q]^{−1}. The estimate (iv) gives

0 ≤ [KL : Q]^{−1} · 4(Ad_{v} + 1) ≤ \frac{16}{[KL : Q]} .

\underline{Lemma 19}: In the situation of lemma 18 we have, that

ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)ζ^{S}(Ad(π), χ ∘ Norm_{K}, s)

and L^{S}(π × π^{∗} ⊗ χ ∘ Norm_{K})L^{S}(σ(π) × π^{∗} ⊗ χ ∘ Norm_{K}) have the same order n at s = 1, a

number independent from χ ∈ Δ̂.

\underline{Proof}: Combine lemma 18 with proposition 1 and 3.

Lemma 19 implies, that the pole number n ≥ 0 defined in lemma 18 is controlled by the order at s = 1 of L^{S}(π × π^{∗}, s)L^{S}(σ(π) × π^{∗}, s). This follows from lemma 18 and 19 with the choice χ = 1.

If π is a cusp form and K is a field, then L^{S}(π × π^{∗}, s) has a simple pole at s = 1. On the other hand L^{S}(σ(π) × π^{∗}, s) has a pole at s = 1 in this case iff σ(π) ≅ π. In case 3 or 1a (if Π is nondegenerate) this is impossible by proposition 2 and therefore n = 1.

If K is not a field but π still cuspidal, then there is a pole of order ≥ 2.

If π is not a \underline{cusp} \underline{form} and K is a field, then the order of L^{S}(π×π^{∗}, s) is at least two. If π is

defined as constituent of Ind(χ_{1}, χ_{2}), then L^{S}(π × π^{∗}, s) = L_{K}^{S}(χ_{1}/χ_{2}, s)L_{K}^{S}(χ_{2}/χ_{1}, s)ζ_{K}^{S}(s)^{2}.

This has order 4 or 2 depending on whether χ_{1} = χ_{2} or not. The other factor is L^{S}(σ(π)×

π^{∗}, s) is L_{K}^{S}(σ(χ_{1})/χ_{2}, s)L_{K}^{S}(σ(χ_{2})/χ_{1}, s)L_{K}^{S}(σ(χ_{1})/χ_{1}, s)L_{K}^{S}(σ(χ_{2})/χ_{2}, s). On the other hand

σ(χ_{1})σ(χ_{2}) = σ(ω_{π}) = ω_{π} = χ_{1}χ_{2}, because ω_{π} = ω_{Π} ∘ Norm_{K}. This implies that the second factor has pole order either 0,2 or 4 at s = 1. Hence n = 2 or n = 4 holds in this case. Remember that n ≤ 4 by lemma 18, because [KL : Q] ≥ 4 holds by assumption.

The case π not cuspidal and K split will not be needed. Nevertheless π is then a constituent of Ind(χ_{1}, χ_{2}) × π_{2}. Then L(π × π^{∗}, s) has a pole of order ≥ 3 at s = 1.

This implies

\underline{Lemma 20}: In the situation of lemma 18 we always get n ≥ 1. The pole number n is equal to n = 1 iff π is cuspidal and K is a field and σ(π) ≅ π̸ (the latter is automatic if Π is non degenerate). In all other cases n ≥ 2. If K is a field, the only possibilities are

n = 2 or n = 4 if n ≠ 1.

\underline{Lemma 21}: In the situation of lemma 18 we have

ord_{s=1} ∏_{χ∈Δ̂} L^{S}(π × π^{∗} ⊗ χ ∘ Norm_{K})L^{S}(σ(π) × π^{∗} ⊗ χ ∘ Norm_{K}) = D · n .

\underline{Proof}: This is the average over χ using lemma 18 and 19.

\underline{Lemma 22}: Let the situation be as in lemma 18. Suppose in addition that KL = L. Then

ord_{s=1} ∏_{χ∈Δ̂} ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) = \frac{D}{2} n − 2 .

\underline{Proof}: Under the assumption KL = L we get from averaging over χ ∈ Δ̂ in lemma 9 and from lemma 18, that

ord_{s=1} ∏_{χ∈Δ̂} ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s) = \frac{D}{2} n + 2 .

Combined with lemma 21 and lemma 18, the proof is completed. The last formula also shows

\underline{Lemma 23}: Let the situation be as in lemma 18. Suppose in addition that KL = L. Then there are at least \frac{D}{4} n + 1 ≥ n + 1 characters χ ∈ Δ̂ for which ζ^{S}(Π, χ, s) has a pole at s = 1.

Remember, that χ_{K} = χ_{0}χ_{T}, where χ_{0} was a character giving rise to a pole (Theorem 2). Especially if χ_{K} = 1 hence K = Q^{2} and KL = L. Then lemma 20 and lemma 22 imply that beside our previous choice of χ_{0}, we can find another choice. As χ_{T} can be taken fixed χ_{K} ≠ 1 can be assumed without restriction of generality.

\underline{Corollary}: For a suitable choice of χ_{0} in theorem 2 we can assume K to be a field.

We can and will assume this from now on \underline{Assumption}: K is a field.

Under this assumption there are only the possibilities n = 1, 2, 4 by lemma 20.

\underline{Final Discussion of case 3 and 1a}:

We use the same notations as in the previous section. We additionally assume that we are in the situation of lemma 3 and lemma 4 (with G = \overline{G}), therefore Π is D-critical with D ≥ 4 of two-abelian type and Π is non degenerate. We separately discuss the four possible pole orders n = 4, 2, 1.

\underline{The case n = 4}:

K is a field and π is Eisenstein attached to χ_{1}, χ_{2}. We show, that this case does not occur.

Lemma 18 would imply ord_{s=1}ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) = 4 for all χ ∈ Δ̂.

For χ = 1 this gives ord_{s=1}ζ^{S}(Ad(π), 1, s) ≥ 2, because ζ^{S}(Π, χ, s) has at most simple poles at s = 1. Thus χ_{2}/χ_{1} = 1.

The group Δ̂ has at least D ≥ 4 elements. If we repeat the same argument for χ ≠ 1, χ_{K} in Δ̂, we get on the other hand (χ_{1}/χ_{2})χ∘Norm_{K} = 1. This together with χ_{2}/χ_{1} = 1 gives a contradiction.

\underline{The case n = 2}:

We assume K to be a field. Recall that Π is nondegenerate, hence π is not cuspidal. This follows from lemma 20. We already know from the discussion preceeding lemma 20, that π is a constituent of Ind(χ_{1}, χ_{2}) with χ_{1} ≠ χ_{2} and σ(χ_{1}) ≠ χ_{1} and σ(χ_{1}) ≠ χ_{2}.

First suppose LK ≠ L and D ≥ 4. Then lemma 21 implies χ_{1}/χ_{2} = χ ∘ Norm_{K} for some

χ ∈ Δ̂. I claim χ^{2} = 1 or equivalently (χ ∘ Norm_{K})^{2} = 1, hence χ_{2}/χ_{1} = χ_{1}/χ_{2} = χ ∘ Norm_{K}.

If χ is not quadratic neither ζ^{S}(Π, χ, s) nor ζ^{S}(Π, χχ_{K}, s) can have a pole at s = 1 by the remark after lemma 10. The product of these two factors is not zero by proposition 1. Then ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) must have a double pole at s = 1 by lemma 18. This is possible only if χ_{2}/χ_{1} = χ_{1}/χ_{2}. This however shows that χ ∘ Norm_{K}, hence also χ, is quadratic.

This being said we see that χ_{2} = χ_{1} · χ ∘ Norm_{K} and χ_{1} = χ_{2} · χ ∘ Norm_{K}. This implies π_{v} ≅ π_{v} ⊗ χ ∘ Norm_{K} for almost all v and contradicts lemma 15.

Therefore we are reduced to the case LK = L and D ≥ 4. Then lemma 22 implies that ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) has a pole at s = 1 for D − 2 elements χ ∈ Δ̂.This allows only the cases D = 8 or D = 4.

If D = 8, then lemma 21 implies that χ_{1}/χ_{2} = χ ∘ Norm_{K} for some χ ∈ Δ̂. If χ_{1}/χ_{2} were not quadratic lemma 18 and n = 2 again shows, that ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) must have a double pole at s = 1. This forces χ_{2}/χ_{1} = χ_{1}/χ_{2}. Therefore χ_{2} = χ_{1}χ ∘ Norm_{K} and χ_{1} = χ_{2}χ ∘ Norm_{K}, which contradicts lemma 15.

Hence only the case LK = L, D = 4 remains. Let χ ∈ Δ̂ be different from 1, χ_{K}. From

lemma 21 then follows χ_{1}/χ_{2} = χ ∘ Norm_{K} or σ(χ_{1})/χ_{1} = χ ∘ Norm_{K} or σ(χ_{1})/χ_{2} =

χ∘Norm_{K}. The first case is impossible by lemma 15, because χ∘Norm_{K} is the nontrivial quadratic character attached to the quadratic field extension L/K. Then lemma 19 shows that ζ^{S}(Π, χ, s) and ζ^{S}(Π, χχ_{K}, s) both have a pole at s = 1. Furthermore lemma 19 shows that either ζ^{S}(Π, 1, s) or ζ^{S}(Π, χ_{K}, s) gives rise to a third pole at s = 1. According to lemma 22 (proof) there are exactly three such poles.

\underline{The case n = 1}:

In this case π is cuspidal and σ(π) ≅ π̸. We maintain the assumption, that K is a field.

If D ≥ 8 or D = 4 and LK ≠ L, then σ(π) is not isomorphic to π ⊗ χ ∘ Norm_{K} for all χ ∈ Δ̂ by proposition 2. Therefore lemma 21 implies

D = n · D = |M_{K}(π)| ,

where M_{K}(π) is the subgroup {χ ∈ Δ̂ | π ≅ π ⊗ χ^{−1} ∘ Norm_{K}} of Δ̂. The group M(π) of characters χ ∈ A A_{K}^{∗}/K^{∗} with π ≅ π ⊗χ has either 1,2 or 4 elements. In our case it contains only one element by lemma 15. We have

0 → Kern(Norm_{K}) → M_{K}(π) → M(π) .

The kernel Kern(Norm_{K}) is trivial if KL ≠ L and {1, χ_{K}} if KL = L. We therefore get D = nD = 2, which contradicts the assumption D ≥ 4. Therefore KL = L and D = 4. From lemma 15 we know π ≅ π̸ ⊗ χ for all characters χ ≠ 1 of A A_{K}^{∗}/K^{∗}. Therefore lemma 19 or 21 implies

σ(π) ≅ π ⊗ χ ∘ Norm_{K} , χ ∈ Ĝ, χ ≠ 1, χ_{K} .

Lemma 22 implies that ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) has no poles at s = 1 for χ ∈ Δ̂. Lemma 18 implies that ζ^{S}(Π, χ, s) has a pole at s = 1 for for exactly one of the two characters χ_{0} ∈ {1, χ_{K}} and exactly one of the two characters χ′_{0} ∈ {χ, χχ_{K}}, where χ ∈ Δ̂, χ ≠ 1, χ_{K}.

Remember χ_{0}χ_{T} , χ′_{0}χ_{T} give rise to χ_{K′} as in theorem 2. In our case we have therefore two choices K_{1}, K_{2} for the algebra K′. For these two choices we find χ_{K_{1}}/χ_{K_{2}} = χ_{0}/χ′_{0}, which is contained in {χ, χχ_{K}}.

We now would like to vary K! For this note, that π = π_{K} and n = n_{K} might depend on the choice of χ_{K} = χ_{T} χ_{0}. However our discusion of the cases n = 4, 2, 1 so far has shown

\underline{Corollary 1}: Given a D-critical Π arising from case 3 or 1a, then [L : Q] = 4. For every choice of χ_{T} as in theorem 2 and any choice of a character χ_{0}, such that

ord_{s=1}ζ^{S}(Π, χ_{0}, s) > 0 ,

we get an algebra K defined by the quadratic character χ_{K} = χ_{T} χ_{0}. If K is a field, then π_{K} is cuspidal and n_{K} = 1 resp. π_{K} is Eisenstein and n_{K} = 2. Whether the first case n_{K} = 1 resp. second case n_{K} = 2 occurs depends on the number ord_{s=1} ∏_{χ∈Δ̂} ζ^{S}(Π, χ, s), which is either 2 in the first case or 3 in the second. This depends only on Π and not on the choices made.

This corollary implies, that we can assume that one of the K_{i} was our fixed choice of field K_{1} = K. For the other choice K_{2} = K′ ≠ K we therefore get, that χ_{K′} ∈ {χ, χχ_{K}}. Especially this second choice is again attached to a nontrivial character χ_{K′}, hence K′ is a field. By construction χ_{K′} ∈ Δ̂. Therefore K′ is contained in L. Because K′ ≠ K and D = [L : Q] = 4 it follows that L = KK′. The three quadratic subfields of L are K and K′ and a third field K′′. It follows that χ_{T} is either 1 or χ_{K′′}. This completes the discussion of the case n = 1.

Remember, that in the case n = 2, there are three different characters χ_{0} in Δ̂, for which ζ^{S}(Π, χ_{0}, s) has a pole at s = 1. These give rise to 3 algebras K_{i} with K_{i} ⊂ if K_{i} is a field. So there are again two choices K_{1} = K and K_{2} = K′ ≠ K, where K, K′ are fields. Outside a finite set of places S we have shown σ(π_{K}^{S}) ≅ π_{K}^{S} ⊗ χ_{L/K}. This implies the statement also for the remaining places in S. Thus the following theorem is proven.

\underline{Theorem 4}: D-critical representations Π of GSp(4, A A) arising from case 3 or 1a can only exist for D = 4. In these cases there are two subfields K = K_{1}, K_{2}, K_{1} ≠ K_{2} of L such that the following holds: The restriction of Π to Sp(4, A A) is a lift of an irreducible automorphic representation π_{K} of Gl(2, A A_{K}), where K is a quadratic extension field of Q contained in L. If σ is the nontrivial automorphism of K/Q, then σ(π_{K}) ≅ π̸_{K} but σ(π_{K}) ≅ π_{K} ⊗ χ_{L/K}. For at least two of the characters χ in Δ̂ the function ζ^{S}(Π, χ, s) has a pole at s = 1.

Let us slightly refine this. First consider the cuspidal case n = 1. From lemma 1 we know for χ ∈ Δ̂

log ζ^{S}(Π, χ, s) ∼ \frac{1}{2}log ζ(s) + ∑_{v} ε_{v}χ(p_{v})Ad_{v}p_{v}^{−s} .

Here ε_{v} = −1 for v not split in L and ε_{v} = 1 else.

We know, that ζ^{S}(π_{K}, χ ∘ Norm_{K}, s) is regular at s = 1 for all χ ∈ Δ̂ for both choices K = K_{1} and K_{2}. Thus

∑_{v∈X} Ad_{v}p_{v}^{−s} ∼ 0

for X either the set of Q-places, which split in L or the set of places which split in K_{i}, but not in L. Then by lemma 9 there exists κ such that

∑_{v∈X} Ad_{v}p_{v}^{−s} ∼ κ · log ζ(s) ,

where X is the set of Q-places which remain inert in the third quadratic field K_{3} defined by χ_{K_{3}} = χ_{K_{1}}χ_{K_{2}}. Let ρ ∈ Gal(L/Q) be the generator of Gal(L/K_{3}). Then lemma 1 gives for χ ∈ Δ̂

log ζ^{S}(Π, χ, s) ∼ log ζ(s)·(\frac{1}{2} − χ(ρ) · κ) .

This implies κ = ±1/2. Note that −1/4 ≤ κ ≤ 3/4 holds by property iv) of D-critical representations, therefore κ = 1/2. We therefore get poles for ζ^{S}(Π, χ_{0}, s) at s = 1 exactly for χ_{0} ≠ 1, χ_{K_{3}} or in other words for χ_{0} = χ_{K_{1}}, χ_{K_{2}}. Then {χ_{K_{1}}, χ_{K_{2}}} = χ_{T} {χ_{0}, χ′_{0}} = χ_{T} {χ_{K_{1}}, χ_{K_{2}}}. This implies χ_{T} = 1 or χ_{T} = χ_{K_{3}}.

Now consider the case n = 2 where π is Eisenstein. There are two fields K_{1} and K_{2} as in theorem 4. We know, that ζ^{S}(π_{K}, χ ∘ Norm_{K}, s) is not regular at s = 1. It has a simple pole for χ = 1, χ_{K} and is regular for the other χ ∈ Δ̂. This again holds for both choices K = K_{1} and K_{2}. Thus

∑_{v∈X} Ad_{v}p_{v}^{−s} ∼ α · log ζ(s)

for X either the set of Q-places, which split in L or the set of places which split in K_{i}, but not in L. Then by lemma 9 there exists κ such that

∑_{v∈X} Ad_{v}p_{v}^{−s} ∼ κ · log ζ(s) ,

where X is the set of Q-places which remain inert in the third quadratic field K_{3}, We get the following orders of ζ^{S}(Π, χ, s) at s = 1 from lemma 9: \frac{1}{2} − κ − α for χ = 1, \frac{1}{2} + κ + α for χ = χ_{K_{1}}, \frac{1}{2} + κ + α for χ = χ_{K_{2}} and \frac{1}{2} − κ + 3α for χ = χ_{K_{3}}. Three of these numbers are 1 and one of them is zero. Hence α + κ = \frac{1}{2}. Then 3α − κ = \frac{1}{2} follows. Therefore ζ^{S}(Π, χ, s) has a pole at s = 1 for χ = χ_{K_{1}}, χ_{K_{2}}, χ_{K_{3}} but not for χ = 1. Furthermore χ_{T} {χ_{K_{1}}, χ_{K_{2}}, χ_{K_{3}}} must contain χ_{K_{1}} and χ_{K_{2}}. Therefore either χ_{T} = 1 or χ_{T} = χ_{K_{3}}.

\underline{Corollary 2}: In the situation and notation be as in theorem 4. If T is regular and chosen as in theorem 2, then χ_{T} is either 1 or χ_{K_{1}}χ_{K_{2}}. Furthermore one of the following two possibilities holds: π_{K_{i}} is cuspidal (for i = 1, 2) and ord_{s=1}ζ^{S}(Π, χ, s) = 1 for exactly two of the four characters χ ∈ Δ̂, namely χ = χ_{K_{1}} and χ = χ_{K_{2}}. Or π_{K_{i}} is Eisenstein (for i = 1, 2) and ord_{s=1}ζ^{S}(Π, χ, s) = 1 for the three nontrivial characters in χ ∈ Δ̂ and

ord_{s=1}ζ^{S}(Π, 1, s) = 0.

\underline{Discussion of case 1b}:

In this section we will give the postponed proof of lemma 10 in the case 1b together with its subsequent consequences.

\underline{Lemma 24}: In case 1b of Taylor’s list the group Q ≅ π_{0}(\overline{G}) is two-abelian of order D = 4. Furthermore Δ ≅ π_{0}(G) is isomorphic to (ZZ/2ZZ)^{3} or (ZZ/4ZZ) × (ZZ/2ZZ) or D_{8}.

\underline{Proof}: Consider the nontrivial quadratic character χ_{Q} of G, which is trivial on \overline{G}. View it as a Dirichlet character of A A^{∗}/Q^{∗}. To χ_{Q} belongs a quadratic extension field L_{Q} of Q. Let L be the field extension of Q attached to s with Galois group Δ of order 2D. It contains L_{Q} with Gal(L : L_{Q}) = Q. Then lemma 7 implies for s → +1

logζ^{S}(Π, 1, s)ζ^{S}(Π, χ_{Q}, s) ∼ 2 · ∑_{v∈X_{1}} −Ad_{v}p_{v}^{−s} + 2 · ∑_{v∈X_{2}}(Ad_{v} + 2)p_{v}^{−s} .

Here X_{2} are the Q-places, which split in L and X_{1} is the set of Q-places which split in L_{Q} but not in L. Note that by lemma 8 χ^{2}/n is attached to a Grossencharacter of A A_{L_{Q}}^{∗}/L_{Q}^{∗}, which does not have finite order. Therefore in the sums above Ad_{v} can be replaced by 1. Thus by Tchebotarev’s theorem for s → +1

logζ^{S}(Π, 1, s)ζ^{S}(Π, χ_{Q}, s) ∼ 2 · (−\frac{D − 1}{2D} + \frac{3}{2D}) · logζ(s) = (\frac{4}{D} − 1) · logζ(s) .

This is possible only for D = 4, because (4/D) − 1 has to be an integer. This proves lemma 10 and lemma 24.

Fix some one dimensional character χ of the group Δ. The Ramanujan conjecture (for a twist of Π) holds for all v ∉ S. Therefore ζ_{v}(Π_{v}, 1, s) = P_{v}(p_{v}^{−s})^{−1} for a real polynomial

P_{v}(T) ∈ R[T]. Then for s → +1

logζ^{S}(Π, χ, s) ∼ ∑_{i} χ(F rob_{v}) ∑_{F rob_{v}∈X_{i}} w_{i}p_{v}^{−s} ,

where i runs over (certain unions of) conjugacy classes X_{i} in Δ. The weights w_{i} are well defined on the conjugacy classes of the elements in Q (cases A and B with weights 3 and -1) and the conjugacy classes of elements in g ∈ Δ \ Q, which are of order 4 (case C with weight −1). The other cases D and E define a union of conjugacy classes with a single weight w ∈ R. It is clear from our discussion of the cases D and E that this weight satisfies −3 ≤ w ≤ 1. In the dihedral case this is a single conjugacy class. In the cases, where the group is abelian, we can use character twists in order to get well defined weights for all classes.

Let us exclude the following case.

\underline{The case Δ = (ZZ/4ZZ) × (ZZ/2ZZ)}: Consider generators N, S with N^{4} = 1, S^{2} = 1. Then Q = {1, N^{2}, S, SN^{2}}. Let χ_{N} be the character of Δ defined by χ_{N}(S) = 1, χ_{N}(N) = i. Note that χ_{N} has imaginary values for all g ∉ Q. Hence for s → +1, s ∈ R

Re(logζ^{S}(Π, χ_{N}, s)) ∼ (3/8 − 1/8χ_{N}(N^{2}) − 1/8χ_{N}(S) − 1/8χ_{N}(SN^{2})) · logζ(s) = 2/8 · logζ(s) .

Note that 2/8 ∉ ZZ, therefore Δ = (ZZ/4ZZ) × (ZZ/2ZZ) is impossible.

\underline{The dihedral case Δ = D_{8}}:

Classes  1 {N^{2}} {S, SN^{2}} {N, N^{3}} {SN, SN^{3}}
Cases  A  B  B  C  D+E
Weights  3  -1  -1  -1  w
χ = 1  1  1  1  1  1
χ = χ_{Q}  1  1  1  -1  -1
χ = χ_{P}  1  1  -1  -1  1
χ = χ_{R}  1  1  -1  1  -1

\underline{Claim}: We have w = 1 and outside a set of density zero in the class determined by

{SN, SN^{3}} the case D is realised. The orders n(χ) = ord_{s=1}ζ^{S}(Π, χ, s) for χ = 1, χ_{Q}, χ_{P} , χ_{R}

are 0, 0, 1, 0 respecively.

\underline{Proof}: We have logζ^{S}(Π, χ, s) ∼ n(χ) · logζ(s) for σ → +1 for integers n(χ). If χ is a

character of Δ, our table gives

n(χ) = \frac{1}{8}(3 − χ(N^{2}) − 2χ(S) − 2χ(N) + 2w · χ(SN)) .

This gives

−n(1) = n(χ_{Q}) = n(χ_{R}) = \frac{1}{4}(1 − w) , n(χ_{P}) = \frac{1}{4}(w + 3) .

We know, that −3 ≤ w ≤ 1, hence either w = 1 or w = −3. It therefore remains only to show, that w = −3 is impossible. In the case w = −3 we have ord_{s=1}ζ^{S}(Π, 1, s) = −1 and we have on the other hand a simple pole at s = 1 for ζ^{S}(Π, χ_{Q}, s). In view of the resume of the last section we can apply theorem 1. It implies, that the restriction of Π^{S} to Sp(4) is the lift of a form π of Gl(2, A A_{K}), where K is a rank two algebra over Q determined by χ_{K} = χ_{T} χ_{Q}. Lemma X implies, that the analog of proposition 1 holds, whose proof did not use property ii) nor L/Q abelian. Therefore

ord_{s=1}(ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{K}, s)) = ord_{s=1}L_{K}^{S}(χ ∘ Norm_{K})L_{K}^{S}(σ(π) × π^{∗} ⊗ (χ ∘ Norm_{K}), s) .

Apply this for χ = 1. The zeta function ζ^{S}(Π, χ_{K}, s) has at most a simple pole at s = 1. On the other hand ζ^{S}(Π, 1, s) has a simple zero at s = 1. Therefore the left hand side is regular at s = 1. The right hand side has a pole of order at least one, coming from the first factor L_{K}(1, s). This is a contradiction, hence the case w = −3 is impossible. The claim is proved.

\underline{Some consequences for the dihedral case}:

1) We already know that ζ^{S}(Π, χ, s) has a pole at s = 1. Therefore the restriction of Π to Sp(4, A A) is attached to some automorphic representation π_{K} of Gl(2, A A_{K}), where K is a rank two algebra over Q defined by the character χ_{K} = χ_{T} χ_{P} (theorem 1).

2) We already know ord_{s=1}ζ^{S}(Π, 1, s) = 0. Note that Π satisfies properties, similar to property i), iii) and iv) of a 2D-critical representation. Therefore proposition 1 can be applied. Proposition 1 applied with χ = 1 gives ord_{s=1}ζ^{S}(Π, χ_{K}, s) = 1.

3) This gives an alternative choice of a rank 2 algebra K′ over Q for which theorem 1 can be applied. K′ is attached to χ_{K′} = χ_{T} χ_{K} = χ_{T}^{2} χ_{P} = χ_{P}. Therefore K′ = L_{P} is the quadratic extension field of Q in L attched to χ_{P}.

\underline{Claim}: χ_{T} = 1.

\underline{Proof}: Outside a set of Q-density zero one of the cases A, B, B, C, D must hold. The explicit description of these cases shows ord_{s=1}ζ^{S}(Π, χ, s) = 0 for every quadratic character χ, which is trivial on Δ. On the other hand ord_{s=1}ζ^{S}(Π, 1, s) = 0. Therefore proposition 1 implies ord_{s=1}ζ^{S}(Π, χ_{K}, s) > 0. Therefore χ_{K} = χ_{P} and χ_{T} = 1. The claim is proved.

4) Write K instead of K′. Let σ be the nontrivial automorphism of K/Q. Proposition 1 for χ = 1 then implies σ(π_{K}) ≅ π̸_{K}. Proposition 1 applied to χ = χ_{Q} with χχ_{K} = χ_{R} gives

ord_{s=1}ζ^{S}(Π, χ_{Q}, s)ζ^{S}(Π, χ_{R}, s) = ord_{s=1}L_{K}(χ_{Q} ∘ Norm_{K})L_{K}(σ(π_{K}) × (π_{K}^{∗} ⊗ χ_{Q} ∘ Norm_{K}, s) .

We know, that the left side has orders equal to zero. Hence σ(π_{K}) ≅ π̸ ⊗ χ_{Q} ∘ Norm_{K}.

5) Let Π_{v} ∼ (ν_{v}, ν̃_{v}, μ_{v}, μ̃_{v}) for v ∉ S. The remarks preceeding lemma 6 give for Q-places v, which split in K = L_{P} with extensions w, w′, that (π_{K})_{w} ×(π_{K})_{w′} ∼ (t_{v}ν_{v}, t_{v}μ_{v})×(t_{v}ν̃_{v}, t_{v}μ̃_{v}) for some t_{v}. The additional factors t_{v} arise through the passage from Π to its twist Π′.

7) Let F ⊂ L be the fixed field of SN ∈ Δ. We claim

σ(π_{K}) ≅ π_{K} ⊗ χ_{F/K} .

It is enough to check this on a set of K-places of K-density one. So it is enough to consider places above Q-places v ∉ T, which split in K. Then F rob_{v} ∈ P = {1, N^{2}, SN, SN^{3}} and there are the follwing possibilies: Either F rob_{v} = 1 or F rob_{v} = N^{2} or F rob_{v} ∈ {SN, SN^{3}}, which corresponds to the cases A, B and D. In case A we have Π_{v} ∼ (ν_{v}, ν_{v}, μ_{v}, μ_{v}), in case B we have Π_{v} ∼ (ν_{v}, −ν_{v}, μ_{v}, −μ_{v}) and in case D we have Π_{v} ∼ (ν_{v}, −ν_{v}, −ν_{v}, ν_{v}). On the other hand χ_{F/K}(F rob_{w}) has the values 1 in case A, −1 in case B, and 1 or −1 in case D for any of the two K-places w lying over v. This proves the claim. As a consequence σ(π_{K}) ≅ π_{K} ⊗ χ_{σ(F)/K} and

π_{K} ≅ π_{K} ⊗ χ_{Q} ∘ Norm_{K} .

Further note χ_{Q} ∘ Norm_{K} = χ_{σ(F)/K}χ_{F/K}.

\underline{The case Δ = (ZZ/2ZZ)^{3}}:

Let the subgroup Q of Δ be generated by N, S with N^{2} = S^{2} = 1 and let Δ = ⟨Q, M⟩ with M^{2} = 1. Let χ_{Q} be the nontrivial quadratic character of Δ, which is trivial on the sub group Q. Let χ_{N}, χ_{S} be the characters defined by χ_{N}(N) = −1, χ_{N}(M) = χ_{N}(S) = 1 and

χ_{S}(S) = −1, χ_{S}(N) = χ_{S}(M) = 1. The weights for F rob_{v} = 1, N, S, NS, M, MN, MS, MNS are 3, −1, −1, −1, w_{1}, w_{2}, w_{3}, w_{4} with the corresponding cases A, B, B, B, D + E, D + E, D +

E, D + E. From this follows

ord_{s=1}ζ^{S}(Π, χ, s)ζ^{S}(Π, χχ_{Q}, s) = m(χ)

with m(1) = 0, m(χ_{N}) = 1, m(χ_{S}) = 1, m(χ_{N}χ_{S}) = 1. Therefore by theorem 1, the restriction of Π to Sp(4, A A) is a lift of some automorphic representation π_{K} on Gl(2, A A_{K}).

Let denote n = ord_{s=1}ζ^{S}(Π, 1, s) and let x, y, z denote the orders of ord_{s=1}ζ^{S}(Π, χ, s) for

χ = χ_{N}, χ_{S}, χ_{N}χ_{S}. Then (w_{1} + w_{2} + w_{3} + w_{4})/8 = n and (w_{1} − w_{2} + w_{3} − w_{4})/8 + 1/2 = x and (w_{1} + w_{2} − w_{3} − w_{4})/8 + 1/2 = y and (w_{1} − w_{2} − w_{3} + w_{4})/8 + 1/2 = z. The fact that

−3 ≤ w_{i} ≤ 1 implies n ∈ {−1, 0} and x, y, z ∈ {0, 1}. Suppose n = −1. Proposition 1 gives ord_{s=1}ζ^{S}(Π, 1, s)ζ^{S}(Π, χ_{K}, s) ≥ 1. The order of ζ^{S}(Π, χ_{K}, s) is at most 1. This is a contradiction, hence n = 0. We can then solve the inequalies for w_{i} above and get

0 ≤ x + y + z ≤ 2 and −2 ≤ −x + y − z ≤ 0 and −2 ≤ x − y + z ≤ 0 and −2 ≤ −x − y + z ≤ 0.

Therefore either x = 1, y = 1, z = 0 (without restriction of generality) or x = y = z = 0. In the first case we get w_{1} = 1, w_{2} = 1, w_{3} = 1, w_{4} = −3 and in the second we get w_{1} = −3, w_{2} = 1, w_{3} = 1, w_{4} = 1. Then for all Q-places except from a set of density zero,

the possible types are A, B, B, B, D, D, D, E resp. A, B, B, B, E, D, D, D.

Hence there are exactly three characters χ = χ_{1}, χ_{2}, χ_{2} of Δ, for which ζ^{S}(Π, χ, s) has a pole at s = 1. Furthermore for all other quadratic characters, especially those which vanish on Δ, the order is zero. We have shown

Δ̂ = {1, χ_{Q}, χ_{1}, χ_{1}χ_{Q}, χ_{2}, χ_{2}χ_{Q}, χ_{3}, χ_{3}χ_{Q}} .

\underline{Claim}: χ_{T} = 1.

\underline{Proof}: For three characters χ_{0} one can apply theorem 1. The corresponding K was defined by χ_{K} = χ_{0}χ_{T}. We already know ord_{s=1}ζ^{S}(Π, 1, s) = 0. By proposition 1 this implies ord_{s=1}ζ^{S}(Π, χ_{K}, s) > 0. Because χ_{K} is a quadratic character this implies χ_{K} ∈ {χ_{1}, χ_{2}, χ_{3}}. In other words multiplication by χ_{T}, which is of order two, stabilizes {χ_{1}, χ_{2}, χ_{3}}. This implies χ_{T} = 1. Especially K is a subfield of L for any choice of χ_{0} ∈ {χ_{1}, χ_{2}, χ_{3}}. Therefore (see the proof of lemma 13)

6 = 2ord_{s=1} ∏_{χ∈Δ̂} ζ^{S}(Π, χ, s) = nD/2 + 2 = 4n + 2 ,

hence n = 1 and lemma 13 gives

ord_{s=1} ∏_{χ∈Δ̂} ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) = nD/2 − 2 = 2 .

The two poles of ζ^{S}(Ad(π), χ ∘ Norm_{K}, s) are, say χ = χ_{4}, χ_{4}χ_{K}. Then for χ ∈ Δ̂ and χ ≠ 1, χ_{K}, χ_{4}, χ_{4}χ_{K} we get from lemma 10

σ(π_{K}) ≅ π_{K} ⊗ χ ∘ Norm_{K} .

\underline{Theorem 5}: In case 1b of Taylor’s list the group π_{0}(G) is of order 8 and either elementary two-abelian or dihedral. The restriction the representation Π to Sp(4, A A) is related to an automorphic representation π_{K} of Gl(2, A A_{K}) as in theorem 2, where K is a quadratic extension field K of Q.

i) Let σ denote the nontrivial involution of K/Q, then σ(π_{K}) ≅ π_{K} ⊗ χ ∘ Norm_{K} for some idele class character χ of A A^{∗}/Q^{∗}.

ii) Every nondegenerate T belonging to a nonvanishing Fourier coefficient of Π, as explained before theorem 1, has trivial discriminant χ_{T} = 1.

Appendix: Balanced representations:

Let ρ = ρ_{1} ⊕ ρ_{2} be balanced. We Let in this section Ñ denote a finite group, and let k denote an algebraicly closed field of caracteristic zero.

\underline{Definition}: Consider representations ρ : Ñ → Gl(V) on finite dimensional k vectorspaces V. We say that ρ has ”only two eigenvalues” if for all g ∈ Ñ the matrix ρ(g) has at most two different eigenvalues ζ_{1}(g), ζ_{2}(g). Let their multiplicities be a_{1}(g), a_{2}(g), with the convention ζ_{1}(g) = ζ_{2}(g) if only one eigenvalue occurs. We put ζ(g) = ζ_{1}(g)/ζ_{2}(g) (well defined in k^{∗} up to inverse).

Let ρ be a representation with only two eigenvalues. Let

\overline{ρ} : Ñ → P Gl(V)

be the associated projective representation. Let Z(ρ) be the kernel of \overline{ρ}. Then ρ|Z(ρ) = χ · id for some character of χ of Z. Let D(ρ) denote the number of elements of Q(ρ) = Ñ/Z(ρ). We have g ∈ Z(ρ) iff ζ(ρ, g) = 1 and the order of root of unity ζ(ρ, g) is the order of the image of g in Q(ρ). If ρ is fixed, we only write Z, Q, D instead of Z(ρ), Q(ρ), D(ρ).

Suppose ρ ≅ ⨁_{j} ρ_{j} decomposes into irreducible representations ρ_{j}. If ρ has only two eigenvalues, then also the representations ρ_{j}. The possible irreducible representations ρ_{j} with only two eigenvalues were classified in [T2], lemma 9 and corollary 1. There are four possible types of representations ρ_{j}

1. dim(ρ_{j}) = 1.

2. dim(ρ_{j}) = 2 and either ρ_{j} is dihedral (induced from a character of a subgroup of index two) or the image Q_{j} = Q(ρ_{j}) of Ñ in the associated two dimensional projective group P Gl(2, k) is A_{4} or S_{4} or A_{5}.

3. dim(ρ_{j}) = d_{j} ≥ 4. In this case there is a subgroup N_{j} of N of index ≥ 16. The quotient Q_{j} = N/N_{j} is an elementary abelian two group, N_{j} acts by scalars under ρ_{j}, and the elements of N \ N_{j} have trace zero and eigenvalues of the form

ζ_{1}(n) = −ζ_{2}(n). Thus ζ(ρ_{j}, n) = 1 resp. ζ(ρ_{j}, n) = −1 if n ∈ N_{j} resp. n ∉ N_{j}.

\underline{Definition}: A representation ρ with only two eigenvalues is called \underline{balanced} if d = dim(ρ) is even and a_{1}(g) = a_{2}(g) = d/2 holds for all g ≠ 1 in Q(ρ). Then formally put

a_{1}(g) = a_{2}(g) also for g = 1 in Q(ρ).

Our main result is

\underline{Lemma 25}: Suppose ρ is balanced. Then ρ is an isotypic multiple of a two dimensional representation, or the associated group Q(ρ) is two-abelian of order D(ρ) ≥ 4.

\underline{Proof}:

\underline{Claim 1}: Let ρ : Ñ → Gl(V) be balanced of dimension d. Let Z be the kernel of \overline{ρ}. Then ρ|Z = χ · id for some character of χ of Z. Suppose (Z, χ) is fixed at the moment. Let ρ′ be a finite dimensional irreducible representation of Ñ of dimension d′, such that ρ′|Z = χ · id. Then there exists an integer 1/c(Q) depending only on Q = Ñ/Z, such that the multiplicity m(ρ, ρ′), with which ρ′ appears in ρ is either zero or

m(ρ, ρ′) = c(Q)dd′ .

If ρ ≅ ⨁ m(ρ, ρ_{i})ρ_{i} is a decomposition into irreducible representations ρ_{i} of dimensions d_{i}, then

∑_{i} d_{i}^{2} = 1/c(Q) .

\underline{Proof}: The multiplicity m(ρ, ρ′) = D^{−1} ∑_{g∈Q} χ_{ρ}(g)\overline{χ}_{ρ′}(g) is zero unless ρ′ appears in ρ.

Then ρ′ satisfies χ_{ρ′}(g) = a′_{1}(g)ζ_{1}(g) + a′_{2}(g)ζ_{2}(g) , a′_{1}(g) + a′_{2}(g) = d′ and ρ is balanced, hence χ_{ρ}(g) = (d/2)(ζ_{1}(g) + ζ_{2}(g)). We get

m(ρ, ρ′) = \frac{1}{D} ∑_{g∈Q}(a_{1}(g)a′_{1}(g) + a_{2}(g)a′_{2}(g) + a_{1}(g)a′_{2}(g)ζ(g) + a_{2}(g)a′_{1}(g)ζ(g)^{−1}) .

= \frac{1}{D} ∑_{g∈Q}(a_{1}(g) + a_{2}(g))(a_{1}(g)′ + a′_{2}(g)) + a_{1}(g)a′_{2}(g)(ζ(g) − 1) + a_{2}(g)a′_{1}(g)(ζ(g)^{−1} − 1)) .

= dd′ + \frac{1}{D} ∑_{g∈Q} a_{1}(g)a′_{2}(g)(s(ord(g)) − 1) + a_{2}(g)a′_{1}(g)(s(ord(g)) − 1))

= dd′ − \frac{dd′}{2D} ∑_{g∈Q}(1 − s(ord(g)))

= dd′c(Q) .

Here ord(g) means the order of the element g in Q. The multiplicative function s is defined in [T2], p.298. For at least one irreducible representation ρ′ the multiplicity m(ρ, ρ′) is nonzero, thus c(Q) > 0.

Decompose ρ ≅ ⨁_{t=1}^{t} m(ρ, ρ_{i})ρ_{i} into irreducible non isomorphic constituents ρ_{i}. Then

m(ρ, ρ_{i})c(Q)^{−1} = dd_{i} and ∑_{i} m(ρ, ρ_{i})d_{i} = d, hence ∑_{i=1}^{t} d_{i}^{2} = 1/c(Q). Especially c(Q)^{−1} is

an integer. Claim 1 is proved.

\underline{Claim 2}: Let ρ be a representation with only two eigenvalues. Put Q = Q(ρ) and D = D(ρ) and

c(Q) = 1 − d(Q) , d(Q) = \frac{1}{2D(ρ)} ∑_{g∈Q}(1 − s(ord(g))) .

Let n_{i}(Q) denote the numbers of elements in Q of order i. Then

a) D/2 ≥ n_{2}(Q) implies c(Q) > 1/8.

b) 3D/4 ≥ n_{2}(Q) > D/2 implies c(Q) > 1/16. If Q is its 2-Sylow group it implies c(Q) >

1/8.

c) n_{2}(Q) > 3D/4 implies Q ≅ (ZZ/2ZZ)^{r} and c(Q) = \frac{1}{D}.

\underline{The proof of a)-c)}: The first statement of c) is Q ≅ (ZZ/2ZZ)^{r} and follow via easy group theory from the knowledge of the number n_{2}(Q) of elements of order two. See [T2]. For g ∈ Q, g^{2} = 1 any h ∈ Q, h^{2} = (gh)^{2} = 1 is in the centralizer Z_{Q}(g) of g. In case c) therefore Z_{Q}(g) has more than D/2 elements. Therefore Z_{Q}(g) = Q. Therefore g ∈ Z(Q) and the centre has more than 3D/4 elements, hence Q = Z(Q).

The statements concerning c(Q) follow from the definition of the function s, as given in [T2], p.298. Namely c(Q) = 1 − d(Q) and 2Dd(Q) = ∑ n_{i}(Q)(1 − s(i)), where s is a weakly multiplicative function with s(p^{n}) = 1, = 1/(1 − p), = 0 if n = 0, 1, ≥ 2. Therefore s(1) = 0

and s(2) = 2 and s(i) ≤ 3/2 for i ≥ 3 and

d(Q) ≤ (2D)^{−1}(2n_{2}(Q) + \frac{3}{2}(D − 1 − n_{2}(Q))) ≤ \frac{n_{2}(Q)}{4D} + \frac{3}{4}(1 − D^{−1}) .

If n_{2}(Q) ≤ D/2, then d(Q) < 7/8. If n_{2}(Q) ≤ 3D/4, then d(Q) ≤ 3/16 + 3/4 − 3/4D < 15/16.

Thus c(Q) > 1/16. If n_{2}(Q) > 1/2 and Q is its 2-Sylow group n_{i}(Q) ≠ 0 only if i is a power of 2 and

d(Q) = (2D)^{−1}(2n_{2}(Q) + ∑_{i>2} n_{i}(Q)) = (2D)^{−1}(n_{2}(Q) + D − 1) < \frac{n_{2}(Q)}{2D} + \frac{1}{2} .

For n_{2}(Q) ≤ 3D/4 this implies d(Q) < 7/8. If n_{2}(Q) > 3D/4, then Q is elementary twoabelian. Hence n_{i}(Q) = 0 for i > 2 and d(Q) = (D − 1)/D. Claim 2 follows.

\underline{Claim 3}: Suppose ρ is balanced and χ_{1} ⊕ χ_{2} ⊕ χ_{3} ↪ ρ contains three nonisomorphic one dimensional representations ρ_{i} = χ_{i}, i = 1, 2, 3. Then all quotients χ_{i}/χ_{j} are quadratic characters.

\underline{Proof}: By a twist one can assume χ_{1} to be 1. Assume χ_{2} to be a character of order ≥ 3 and χ_{3} to be of order ≥ 2. Because the representations ρ has only two eigenvalues,

χ_{2}(g) ≠ 1 and χ_{3}(g) ≠ 1 implies χ_{2}(g) = χ_{3}(g). This implies χ_{2} = χ_{3}.

\underline{Claim 4}: Suppose ρ is balanced and suppose ρ_{1} ⊕ ρ_{2} ↪ ρ with ρ_{1} irreducible of dimension 2 and ρ_{2} a character. Then Q = Q(ρ_{1}) is an elementary abelian two group.

\underline{Proof}: We can reduce to the case ρ_{2} = 1 by a twist. Then ζ_{1}(g) = 1. Put Q = Q(ρ_{1}) and

Z = Z(ρ_{1}) = Kern(\overline{ρ}_{1}) and k = |Z|. Suppose ρ_{1}|Z = χ. Then

0 = m(1, ρ_{1}) = (kD)^{−1} ∑_{g∈QZ} \overline{χ}_{ρ_{1}}(g)

= (kD)^{−1} ∑_{g∈QZ}(a_{1}(g) + a_{2}(g)ζ(g)^{−1}) + (kD)^{−1} ∑_{h∈Z}(2χ(h) − 2)

= d_{1} − (kD)^{−1} ∑_{g∈ZQ} a_{2}(g)(1 − ζ(g)^{−1}) − \frac{2κ}{D} .

Here κ = 0, 1 depending on whether χ = 1 or not. But d_{1} = 2 implies a_{2}(g) = a_{1}(g) if g ≠ 1 in Q(ρ_{1}), therefore

0 = 2(1 − \frac{1}{2D} ∑(1 − s(ord(g))) − \frac{2κ}{D} = 2c(Q) − \frac{2κ}{D}

or c(Q) = \frac{κ}{D}. This excludes κ = 0, because we already know, that c(Q) > 0. Then c(Q) = 1/D. The rest follows from

\underline{Claim 5}: c(Q) = \frac{1}{D} iff Q is elementary two-abelian.

From [T2] we have c(D) = 1−(D)^{−1}(n_{2} +∑_{i≥3} α_{i}n_{i}) with 0 < α_{i} < 1. Therefore 1/D = c(Q)

implies ∑_{i≥3} α_{i}n_{i} = ∑_{i≥3} n_{i}, hence n_{i} = 0, i ≥ 3. In other words Q is elementary two abelian. The converse is clear.

\underline{Claim 6}: Suppose ρ is balanced and Q = Q(ρ). If Q is elementary two-abelian, then

ζ(g) = −1 if g ≠ 1 in Q(ρ) and ζ(g) = 1 if g = 1 in Q.

\underline{Claim 7}: Let Q = Q(ρ) and ρ be balanced with c(Q) > 1/8. Then ρ is an isotypic multiple ρ = a · ρ_{1} of a two dimensional representation ρ_{1}, or Q is an elementary abelian two-group of order D ≤ 4. We call the second case D-critical. Any 4-critical representation has the

form ρ ≅ χ ⊗ (1 ⊕ χ_{1} ⊕ χ_{2} ⊕ χ_{1}χ_{2}).

\underline{Proof}: Suppose Q is an abelian two-group, then c(Q) = 1/D. Thus c(Q) > 1/8 implies D < 8, hence D ≤ 4. c(Q) > 1/8 implies ∑_{i} d_{i}^{2} ≤ 7 and c(Q)^{−1} ≤ 7 by claim 1. Hence there is at most one i with d_{i} = 2, all others must be d_{j} = 1, j ≠ i. The only possibility in the case d_{1} = 2 is then t ≤ 4.

Suppose ρ is not isotypic multiple of an irreducible two dimensional representation. Then there are two cases: Either ρ is a direct sum of characters (d_{i} = 1), or there is one irreducible two dimensional constituent ρ_{1} with multiplicity 2a and (up two 3 distinct, at least one) characters χ_{i} with multiplicity a each (claim 1). In the first case Q is elementary two abelian by claim 3. The character has to be of the form indicated. Let us show that the second case cannot occur. In the second case consider the group Q_{1} = Q(ρ_{1}). By claim 4 the group Q_{1} must be elementary two abelian. The list of irreducible two dimensional cases of [T2], corollary 1 then implies, that ρ_{1} is dihedral of type D_{2n}, n = 2. By claim 1 we have ρ = 2a · ρ_{1} ⊕ ⨁_{1,..,t} a · χ_{i} (with 1 ≤ t ≤ 3). By a character twist we can assume χ_{1} = 1.

Let Q(ρ) → Q(ρ_{1}) be the natural surjection. Let U be its kernel. For g ∈ U we must have χ_{i}(g) = 1, because ρ is balanced. Otherwise the eigenvalue 1 occurs with multiplicity a_{1}(g) ≥ 4a + a and the second eigenvalue different from 1 occurs only with multiplicity a_{2}(g) ≤ a + a, which is impossible. Therefore χ_{i} are characters of the group Q(ρ_{1}) ≅ D_{4}. The representations a · ρ_{1} and ρ are both balanced representations of D_{4}. Therefore ⨁_{1,..,t} ·χ_{i} must be balanced. Because t ≤ 3 this implies t = 2. But Q(ρ_{1}) has order 4 and then χ_{2}(g) must be −1 for 3 elements g ∈ Q(ρ_{1}) by a comparison with the eigenvalues of D_{4}. An abelian character of this group cannot have this property. This is a contradiction.

\underline{Claim 8}: Suppose ρ is balanced with Q = Q(ρ). Suppose c(Q) ≤ 1/8, then Q is twoabelian.

\underline{Proof}: By claim 2 we can assume c(Q) > 1/16 and n_{2}(Q) > D/2, and Q is not its 2-Sylow group. Therefore d_{i}^{2} < 16 holds by claim 1 and especially d_{i} < 4. This implies by [T2], lemma 9 that d_{i} = 2 or d_{i} = 1.

Let ρ = ⨁_{i} m_{i}ρ_{i} be the decomposition in irreducible constituents. There is a natural map

0 → K → Q(ρ) → ∏_{i} Q(ρ_{i}) ,

whose kernel K is abelian. K is furthermore in the center of Q(ρ).

Suppose one of the ρ_{i} is one dimensional. Then all Q(ρ_{i}) are two-abelian by claim 4. Thus Q(ρ) is a central extension of a two-abelian group by an abelian group K. Therefore Q ≅ Q′ × K′, where Q′ is a 2-group and K′ is odd abelian. Then n_{2}(Q) > D/2 implies K′ = 0 and Q = Q′ is a 2-group. Then claim 2b) shows c(Q) > 1/8, a contradiction. Therefore one dimensional characters ρ_{i} cannot occur. All ρ_{i} must be 2-dimensional. At most three of them occur, each with equal multiplicity, which we can assume to be one.Then claim 1 gives m(ρ, ρ_{i})c(Q)^{−1} = dd_{i}, hence c(Q)^{−1} = dd_{i} = 2d = 2 · 2t, where t = 1, 2, 3. If t = 1, then c(Q) > 1/8 contrary to our assumption. Thus t = 2 or t = 3.

Each of the maps Q(ρ) → Q(ρ_{i}) is surjective. Therefore n_{2}(Q(ρ)) > D(ρ)/2 implies n_{2}(Q(ρ_{i})) > D(ρ_{i})/2. The list in [T2] therefore excludes the case Q(ρ_{i}) = A_{4}, S_{4}, A_{5}. There only remain the (two dimensional irreducible) cases, where Q(ρ_{i}) = D_{2n_{i}} with even n_{i}.

Suppose finally, that ρ is the sum of two (or three) irreducible 2-dimensional representation of dihedral type. Let U ⊂ Q be the normal subgroup of index two belonging to ρ_{1}. Then the restriction of ρ to U is again balanced, but now contains 2 different one dimensional characters. By claim 4 the groups U_{i} are therefore two-abelian. Therefore U is the extension of a two-abelian group by the central abelian group K. Furthermore Q is an extension of a 2-group by a central abelian group K. As above this implies, that Q is a 2-group. Our assumption c(Q) ≤ 1/8 and claim 2b) therefore imply, that we are in the situation of claim 2c with Q two-abelian.

Claim 7 and 8 imply

\underline{Claim 9}: Suppose ρ is balanced. Then ρ is an isotypic multiple of a two dimensional representation, or the associated group Q = Q(ρ) is two-abelian of order D ≥ 4. If D = 4, then ρ is a multiple of χ ⊗ (1 ⊕ χ_{1} ⊕ χ_{2} ⊕ χ_{1}χ_{2}) for characters χ, χ_{1}, χ_{2} with χ_{1}^{2} = χ_{2}^{2} = 1.

References:

[AC] Arthur J., Clozel L., Simple algebras, base change, and the advanced theory of the trace formula, Princeton university press, 1989 [GJ] Gelbart Jaquet

[He] Henniart G., Representations l-adiques abeliennes, Seminaire de theorie des nombres, Paris 1980-81, Seminaire Delange-Pisot-Poitou, Birkhäuser 1982

[HST] Harris M., Soudry D., Taylor R., l-adic representations associated to modular forms over imaginary fields, I. Lifting to GSp(4, Q), Inv. math. 112, 377 - 411 (1993)

[Hu] Huppert B., Endliche Gruppen I, Grundlehren d. math. Wiss. 134, (1983)

[KV] Kashiwara M., Vergne M., On the Segal-Shale-Weil representation and harmonic polynomials, Invent. Math. 44 (1978), 1 - 47

[KRS] Kudla S.S., Rallis S., Soudry D., On the degree 5 L-function for Sp(2), Inv. Math 107, 483 - 541 (1992)

[LL] Labesse J.P., Langlands R., L-indistinguishability for Sl(2), Can. J. Math. , vol XXXI, n. 4, 726 - 785, 1979

[Pr] Przebinda T., , The oscillator correspondence for the pair O(2, 2), Sp(2, R), Mem. Am. Math. Soc. 403 (1989)

[S] Soudry D., The CAP representations of GSp(4), Crelle journal 383, 97 - 108 (1988)

[T] Taylor R., On the l-adic cohomology of Siegel threefolds, Inv. math. 114, 289 - 310 (1993)

[T2] Taylor R., Galois representations associated to Siegel modular forms of low weight, Duke math. journal, vol. 63, n.2, 1991

[V] Vigneras M.F., Correspondance entre represenataions automorphes de Gl(2) sur une extension quadratique de GSp(4) sur Q, conjecture locale de Langlands pour GSp(4), Conteporary Mathematics vol. 53, 1986

CAP-LOCALISATION

Rainer Weissauer

Mannheim

November 1993, revised version

1.Introduction

Let

S_{K}(G) = G(Q) \ (X_{G} × G(A A_{f})/K)

be an arithmetic quotient with respect to a connected reductive group G over Q. The space X_{G} = G_{∞}/Z_{G,∞}^{0}K_{∞} is the connected symmetric space attached to a reductive algebraic group G over Q. K_{∞} denotes a maximal compact subgroup of G_{∞} and K a small compact open subgroup of G(A A_{f}). Furthermore we assume, that G and its Levi subgroups satisfy the strong approximation condition (condition A1 of the appendix).

The cohomology groups H^{ν}(S_{K}(G), E_{λ}) for suitable coefficient systems E_{λ} are modules under the Hecke algebra. The coefficient systems E_{λ} are attached to rational irreducible representations of the group G, with highest weights λ. Suppose K = ∏_{v} K_{v}. Consider a finite set S of places containing all archimedian places, such that for all places v ∉ S the group K_{v} is a special, good maximal compact subgroup of G_{v}. Let

π^{S} = ⊗_{v∉S}π_{v}

be an irreducible admissable spherical representation of G(A A_{f}^{S}) or equivalently, by abuse of notation, the irreducible representation of the spherical Hecke algebra outside S, which it induces. Let

H^{ν}(S_{K}(G), E_{λ})(π^{S})

be the π^{S}-isotypic constituents of the cohomology. These vector spaces are modules under the Hecke algebra H_{S} of locally constant functions with compact support. The main purpose of this note is to derive a simple formula for the trace of f_{S} ∈ H_{S} = ⊗_{v∈S}H_{v} in the subspace of allowable operators H_{S,K} (see appendix 1)

T r_{s}(f_{S}) = ∑_{ν}(−1)^{ν}tr(f_{S}, H^{ν}(S_{K}(G), E_{λ})(π^{S})) .

Our formula is an application of the topological trace formula of Goresky and MacPherson. We will especially show the following. Assume that the unramified spherical representation π^{S} of G(A A_{f}^{S}) is not isomorphic to a subquotient of an induced representation Ind_{P^{S}}^{G^{S}}(σ^{S}), where P ≠ G is parabolic with Levi component L and σ^{S} is an automorphic representation of L(A A_{f}^{S}). Then π^{S} is cuspidal and furthermore π^{S} is not a CAP representation (in the sense of [PS],[S]). Under these assumption we get a very simple formula for the trace of f_{S}. See section 9 and 10.

Of special interest is the case where G_{∞} has discrete series representations. This is discussed in section 11. Then the formula is the following. If the group K is small and π^{S} is not CAP, the trace T r_{s}(f_{S}) of f_{S} equals

d(G) · ∑_{γ∈G(Q)/∼}^{′}[G_{γ} : G_{γ}^{0}]^{−1}τ(G_{γ}^{0})O_{γ}^{G(A A )}(f_{S}f_{π}f_{∞}) .

The sum is over all semisimple R-elliptic conjugacy classes in G(Q), G_{γ}(Q) denotes the centralizer of γ in G. The coefficients O_{γ}^{G(A A )} are orbital integrals. Measures are such that vol_{dg_{f}} (K) = 1 and vol_{dg_{∞}dg_{f}} (G(Q) \ G(A A)) = τ(G) is the Tamagawa number. The function f_{π^{S}} is a suitable projector, a good π^{S}-projector as explained in the text, and f_{∞} = (−1)^{q}d(G)^{−1} ∑ f_{π_{∞}} is defined as a suitable linear combination of pseudo coefficients of discrete series representations with respect to dg_{∞}. The corresponding L-packet is determined by the coefficient system E_{λ}. Here d(G) denotes the number of discrete series representations in the L-packet.

If K_{∞} is replaced by a subgroup of finite index, the similar formula holds but then has to be multiplied by the index.

2.The Cohomological Trace Formula

Goresky and MacPherson [GMP] gave an explicit formula for the supertrace of Hecke operators f on cohomology L(f) = T r(f; H^{•}(S_{K}(G), E_{λ})). They describe their formula in a classical context. Later we reformulate it into an adelic setting.

Let G be a connected reductive group over Q. Assume G satisfies assumption A1 of the appendix. Let A A denote the ring of adeles of Q and A A_{f} the ring of finite adeles. Let K = ∏_{v finite} K_{v} be a small compact open subgroup of G(A A_{f}). The meaning of small will be explained later. It garanties the assumption Γ neat, used by Goresky-MacPherson. Consider Hecke operators. They are linear combination of Hecke operators corresponding to double cosets KgK for g ∈ G(A A_{f}). Assume that f is allowable, i.e. all the (K, g) satisfy (A2 and) A3 as explained in appendix 1. For such choice of K and g the adelic double coset KgK corresponds to a classical Hecke operator attached to the double coset ΓgΓ for some g ∈ G(Q), viewed as an element from G(A A_{f}), as explained in appendix 1. Under these assumptions the trace of the Hecke correspondence ΓgΓ is

L(f) = ∑_{P} ∑_{F} LC(F)

LC(F) = χ(F) · r(γ) · Ψ(γ_{L}, λ)

Ψ(γ_{L}, λ) = ∑_{w∈W′}(−1)^{l(w)}ψ_{w(λ+ρ_{G})−ρ_{G}}(γ_{L}^{−1}) .

See [GMP]. The formulas of Goresky Macpherson remain true for a finite numbers of disjoint copies of symmetric spaces (which arise in the adelic situation) as long as the Hecke operators are classical ones. Namely, if they permute some of the copies, this gives no contribution to the Lefschetz number. Of course the formulas above need some explanations.

The first sum runs over Q-parabolic groups P, those occuring as boundary strata of the reductive Borel-Serre compactification. The second is over all connected components F of the fixed point locus of the Hecke operator attached to ΓgΓ on the given P-strata of the reductive Borel-Serre compactification S_{K}(G)^{+}. Connected components are understood in the topological sense. χ(F) denotes the Euler characteristic of F. To each fixed point component F is attached an element γ ∈ P (Q) such that γ ∈ ΓgΓ. To define γ Goresky and MacPherson consider the group Γ′ = Γ ∩ g^{−1}Γg and lifts of the maps c_{1}(Γ′y) = Γy and

c_{2}(Γ′y) = Γgy of the form d_{1}(y) = γ_{1}y and d_{2}(y) = γ_{2}gy. Then d_{2}d_{1}^{−1}(z) = γ_{2}gγ_{1}^{−1}z = γz,

where γ = γ_{2}gγ_{1}^{−1} ∈ ΓgΓ. The lifts are chosen such that the fixed point equations γ_{1}\overline{p} = \overline{x} and γ_{2}g\overline{p} = \overline{x}, in the notation of [GMP], p.468, hold. This amounts to γ(γ_{1}\overline{p}) = (γ_{1}\overline{p}). The element γ is in P (Q), uniquely determined up to conjugation in Γ ∩ P (Q).

The projection γ_{L} of γ onto the Levi component L(Q) of P (Q) is semisimple and elliptic ([GMP] page 469 and 474). Only those F appear in the first sum, for which the element γ_{L} is P-contractive (I(γ) = ∅ in the sense of [GMP], p.471). This means |γ_{L}^{σ}|_{∞} > 1 for all simple roots σ of a maximal Q-split torus in the center of P on the Liealgebra of the nilpotent radical of P. W′ is a subset of the Weyl group (considered over the algebraic closure). It consists of those elements w, here Φ^{+} ∩wΦ^{−} ⊂ Φ(Lie(N_{P})) ([GMP] page 474). Suppose P = P_{θ} = L_{θ}N_{θ} is a standard parabolic attached to a subset θ of simple roots Δ (considered over the algebraic closure). Then W′ = W^{P_{θ}} is the set of all w ∈ W such that w^{−1}(α) > 0 for all α ∈ θ. This is clear, because Φ(Lie(N_{P})) is the set of those positive roots, which are not linear combinations of simple roots α ∈ θ. Therefore w ∈ W′ means wΦ^{−} ∩ Φ_{L_{θ}}^{+} = ∅. The Weyl group W of G (considered over the algebraic closure) is the disjoint union of the cosets W_{L_{θ}} ·w for w ∈ W^{P_{θ}}. Especially |W^{P_{θ}}| = |W |/|W_{L_{θ}}|. See [BW], p.87. The elements w ∈ W^{P_{θ}} are the unique representatives of minimal lenght in the W_{M_{θ}} left cosets. Finally ψ_{λ} denotes the character of the finite dimensional representation of L with highest weight λ, where λ is a dominant weight of L. Remember, ρ_{G} is half of the sum of the positive roots.

The following inductive property of the sets W^{P} ⊂ W is important. Let P = P_{θ_{1}} ⊂ Q = P_{θ_{2}} ⊂ G be standard parabolic subgroups, corresponding to θ_{1} ⊂ θ_{2}. Let L = L_{θ_{2}} be the standard Levi component of P_{θ_{2}}. Then P_{θ_{1}} ∩ L = P′ is a standard parabolic subgroup of L with Levi component L′ = L_{θ_{1}}. Especially, W^{P′} ⊂ W_{L} is defined and we get

W^{P′} · W^{Q} = W^{P} .

Namely w_{1} · w_{2} = w′_{1} · w′_{2} with w_{1}, w′_{1} ∈ W^{P′} and w_{2}, w′_{2} ∈ W^{Q} implies W_{L_{θ_{2}}} w_{2} = W_{L_{θ_{2}}} w′_{2},

hence w_{2} = w′_{2} and therefore also w_{1} = w′_{1}. By the above mentioned formula for the cardinalities it is enough to show, that the set on the left side is contained in the one on the right. But this is clear. Every w_{1}^{−1} for w_{1} ∈ W^{P′} maps Φ(L_{θ_{1}})^{+} to Φ(L_{θ_{2}})^{+}, every w_{2}^{−1}

for w_{2} ∈ W^{P_{θ_{2}}} maps Φ(L_{θ_{2}})^{+} to Φ^{+} = Φ(G)^{+}. Thus w_{1}w_{2} ∈ W^{P_{θ_{1}}}.

Goresky-MacPherson define Γ_{N} = Γ ∩ N, Γ′ = Γ ∩ γ^{−1}Γγ, Γ′_{N} = Γ′ ∩ N and the numbers

r = r(γ) = [Γ_{N} : Γ′_{N}] = [Γ_{N} : Γ_{N} ∩ γ^{−1}Γ_{N}γ]
s = s(γ) = [γ^{−1}Γ_{N}γ : Γ′_{N}] = [Γ_{N} : γΓ′_{N}γ^{−1}] .

\underline{Lemma 1}: s(γ) = r(γ^{−1}).

\underline{Proof}: s(γ) = [Γ_{N} : γ(Γ_{N} ∩ γ^{−1}Γ_{N}γ)γ^{−1}] = [Γ_{N} : γΓ_{N}γ^{−1} ∩ Γ_{N}] = r(γ^{−1}).

\underline{Lemma 2:} s(γ)/r(γ) = γ^{2ρ_{P}}.

\underline{Proof}: The quotient [Γ_{N} : Γ_{N} ∩ γΓ_{N}γ^{−1}]/[Γ_{N} : Γ_{N} ∩ γ^{−1}Γ_{N}γ] is

[Γ_{N} ∩ γ^{−1}Γ_{N}γ : Γ_{N} ∩ γΓ_{N}γ^{−1}] = [Γ_{N} ∩ γ^{−1}Γ_{N}γ : γ(Γ_{N} ∩ γ^{−1}Γ_{N}γ)γ^{−1}] = |γ^{2ρ_{P}} |_{∞} .

\underline{Remark:} The assumptions A1 and A3 are probably to strong or even unneccessary. They allow to transfer the assertions of [GMP] from the classical to the adelic setting. This will be done in the next sections. Due to the local nature of the Lefschetz fixed point formula they are very likely not needed at all. However, as a complete proof of the announcements of [GMP] has not yet been published, we insert them in order to be able to rely on the assertions of [GMP] alone. In the later part of this paper we will make the assumptions A1 and A3 also for certain subgroups M of the Levi subgroups L and all Heckeoperators which are induced on these subgroups of the Levi subgroups. The precise meaning of this will become clear later. It should be remarked, that this bunch of assumptions means no restriction in the case of split groups like G = GSp(2n, Q) or

Gl(n, Q).

3.The adelic reductive Borel-Serre compactification:

1) As a set the Borel-Serre compacification is a quotient of a union of symmetric spaces X_{L} attached to the Levi components L of all parabolic subgroups P of G defined over Q. Its adelic counterpart is

(S_{K}^{G})^{+} = G(Q) \ [⋃_{P} X_{L} × (G(A A_{f})/K)]

= ⋃_{P_{0}⊂P} P (Q) \ [X_{L} × (G(A A_{f})/K)] .

Observe that G(Q) operates transitively on minimal Q-parabolic groups. If we fix some minimal parabolic P_{0}, the second union is over the finitely many representatives of conjugacy classes of Q-parabolics. We choose them to be the standard parabolic Q-groups containing P_{0}. Remember, the stabilizer of each P under conjugation is P (Q).

2) Put K̃_{∞} = K_{L,∞}Z_{L,∞}^{0}. Here K_{L,∞} denotes a maximal compact subgroup of L_{∞} and Z_{L,∞}^{0} denotes the topological connected component of the center of L_{∞}. Iwasawa decomposition gives X_{L} = L_{∞}/K̃_{∞}, An easy density argument allows to replace N(Q) by N(A A_{f}). Hence

(S_{K}^{G})^{+} = ⋃_{P_{0}⊂P} L_{P} (Q)N_{P} (A A_{f}) \ L_{∞} × G(A A_{f})/K̃_{∞}K .

3) Consider a point \overline{x} ∈ (S_{K}^{G})^{+}. Choose a representative x_{∞}x_{f} ∈ L_{∞} × G(A A_{f}) for \overline{x}. Consider the operator

T (g^{−1}) : x_{∞}x_{f} ↦ x_{∞}x_{f}g^{−1} .

Put K′ = K ∩ g^{−1}Kg. The Hecke correspondence is given by the operators

(S_{K′}^{G})^{+} ⟶^{⟶} (S_{K}^{G})^{+}

by the two maps c_{1} = T (1) and c_{2} = T (g^{−1}). See appendix 1. As explained there, we can assume g ∈ G(Q). Then T (g^{−1}) is the correspondence given by x_{∞} ↦ x_{∞}g^{−1} ≅ g_{∞}x_{∞}. This is exactly the Hecke correspondence considered in [GMP], p.467 attached to the double coset KgK.

4.Cusps:

Consider \underline{cusps} h represented by the double coset of x_{f} in P (Q) \ G(A A_{f})/K. For such h put

Γ_{P_{h}} = P (Q) ∩ x_{f}Kx_{f}^{−1} and Γ_{N_{h}} = N(Q) ∩ x_{f}Kx_{f}^{−1} .

Consider the adelic versions K_{N}(h) = N(A A_{f}) ∩ x_{f}Kx_{f}^{−1} and K′_{N}(h) = N(A A_{f}) ∩ x_{f}K′x_{f}^{−1}. Obviously

[K_{N}(h) : K′_{N}(h)] = [Γ_{N_{h}} : Γ′_{N_{h}}] .

Now assume, that the following fixed point equation γ^{−1}x_{f}g^{−1} = x_{f}k for some γ ∈ P (Q), k ∈ K (studied later) holds. By definition we have gK′g^{−1} ⊂ K. This and the fixed point equation implies γ^{−1}x_{f}K′x_{f}^{−1}γ ⊂ x_{f}Kx_{f}^{−1}. Hence

γ^{−1}Γ′_{P_{h}}γ = P (Q) ∩ γ^{−1}x_{f}K′(γ^{−1}x_{f})^{−1} ⊂ P (Q) ∩ x_{f}Kx_{f}^{−1} = Γ_{P_{h}} .

But x_{f}K′x_{f}^{−1} = x_{f}Kx_{f}^{−1} ∩x_{f}g^{−1}K(x_{f}g^{−1})^{−1} equals x_{f}Kx_{f}^{−1} ∩γ(x_{f}Kx_{f}^{−1})γ^{−1}, by the assump-

tion x_{f}g = γx_{f}k. Intersection with P (Q) gives Γ′_{P_{h}} = Γ_{P_{h}} ∩ γΓ_{P_{h}}γ^{−1}. Therefore, assuming the fixed point equation T (g^{−1})h = h or more precisely x_{f}g^{−1}K = γx_{f}K with γ ∈ P (Q), we get

\underline{Lemma 3}: [K_{N}(h) : K′_{N}(h)] = [Γ_{N_{h}} : (Γ_{N_{h}} ∩ γΓ_{N_{h}}γ^{−1})] = r(γ^{−1}).

5.Small groups:

Consider compact subgroups K ⊂ G(A A_{f}) and K̃_{∞} = K_{∞}Z_{G,∞}^{0}. Here K_{∞} is maximal compact in G_{∞}. K will be called \underline{small}, if

x^{−1}nγx ∈ KK̃_{∞}Z_{L,∞}

for x ∈ G(A A), n ∈ N(A A), γ ∈ P (Q) and any Q-parabolic P = L · N with unipotent radical N implies γ_{L} ∈ Z_{L}(Q) (image in the Levi component is contained in the center) and furthermore γ_{L} = 1, if γ_{L} is a torsion element.

\underline{Remark}: Of course it is enough to demand this for all standard parabolic groups containing a fixed P_{0}.

\underline{Remark}: Small implies neat in the sense, that L(Q)_{tor} ∩ (xKx^{−1} ∩ P)_{L} = 1.

Small groups exist according to the following considerations. There are only finitely many x ∈ P (A A) \ G(A A)/K for any K of finite index in a maximal compact subgroup of G(A A). Conjugation under x replaces K by K_{k} and gives equations p^{−1}nγp ∈ K_{k} for p ∈ P (A A) instead. This is equivalent to m^{−1}γ_{L}m ∈ (K_{k} ∩ P)_{L} for m ∈ L(A A). The index with respect to L indicates projection from P to the Levi component L. But γ_{L} is automatically semisimple, because it lies in a maximal compact subgroup of L_{∞} (modulo the center). L and L/Z(L) are connected reductive groups. Hence the assumption of Kottwitz [K], prop. 8.2 are satisfied. Using the argument of [K], prop. 8.2 such equations can hold only for finitely many L(\overline{Q}) conjugacy classes of elements γ_{L} in L(Q) modulo the center. Shrinking K leaves us, using considerations on eigenvalues, with the unique \overline{Q} conjugacy class {1} mod Z_{L}(Q). Thus γ_{L} ∈ Z_{L}(Q). But Z_{L} is a diagonalizable group. For such groups Z and any compact subgroup K of Z(A A_{f}) the intersection Z(Q)_{tor} ∩ K is finite. Further shrinking of K therefore forces (Z_{L}(Q))_{tor} ∩ K = 1 for all the finitely many cases to be considered.

\underline{Remark}: Our assumption A1 on the center Z_{G} of G implies that Z_{L}(Q)∩K ⊂ Z_{L}(Q)_{tor} ∩K is finite.

6.Fixed point sets:

We want to determine the fixed points of T (g^{−1}) on a given P-stratum of the Borel-Serre compactification (S_{K}^{G})^{+}. For that purpose we fix some Q-parabolic group P = L_{P} N.

1) Let x ∈ (S_{K′}^{G})^{+} be a fixed point in in the P-stratum, such that T (1)(x) = T (g^{−1})(x) holds in (S_{K}^{G})^{+}. In terms of representatives x = x_{f}x_{∞} for x this means

γ^{−1}x_{∞}x_{f}g^{−1} = x_{∞}x_{f} · k^{−1}

for certain γ ∈ P (Q) and k ∈ K̃_{∞}K. Equivalently this can be written in the form

x^{−1}γx ∈ g^{−1}KK̃_{∞} .

In the special case x_{f} = 1 and x = x_{∞} this means

d_{2}d_{1}^{−1}(xK̃_{∞}) = d_{2}(xK_{∞}) = (xK̃_{∞})g^{−1} = γ_{∞}xkK̃_{∞},

or d_{2}d_{1}^{−1}(x_{∞}K̃_{∞}) = (γ)_{∞}(x_{∞}K̃_{∞}). Hence γ ∈ P (Q) is the element defined by Goresky and MacPherson ([GMP], p.469).

2) Now consider the full \underline{conjugacy class} of a fixed element γ in the group P (Q). Let us determine the the set of fixed points F ix(γ) in the P-stratum L_{P} (Q)N(A A_{f}) \ L_{∞} × G(A A_{f})/K̃_{∞}K′ of the reductive Borel-Serre compactification (S_{K}^{G})^{+}, belonging to some of the P (Q)-conjugates of γ. Hence

F ix(γ) ⊂ L_{P} (Q) \ ((L_{∞}/K̃_{∞}) × (N(A A_{f}) \ G(A A_{f})/K′)) .

In terms of representatives F ix(γ) is the set of all double coset with representatives x = x_{∞}x_{f} ∈ L_{∞} × G(A A_{f}) such that there exist n ∈ N(A A_{f}), δ ∈ P (Q), k ∈ K̃_{∞}K satisfying

n(δ^{−1}γ^{−1}δ)x_{∞}x_{f}g^{−1} = x_{∞}x_{f}k .

This is equivalent to

(∗) x^{−1}nδγδ^{−1}x ∈ g^{−1} · K̃_{∞}K

for some n ∈ N(A A_{f}). By abuse of notation γ, δ denote the elements in G(Q) and also their images in G_{v} or G(A A_{f}).

\underline{Remark}: The equation y_{∞}^{−1}γ_{L}y_{∞} ∈ K̃_{∞} for y_{∞} = δx_{∞} implies, that the element γ_{L} ∈ L(Q) is R-elliptic, hence automatically semisimple.

3) Consider the sets of local solutions: For nonarchimedian places v

Sol_{v}(γ) = {x_{v} ∈ N(Q_{v}) \ G(Q_{v})/K′_{v} | for some n ∈ N_{v} holds x_{v}^{−1}nγx_{v} ∈ g_{v}^{−1}K_{v}} ,

where K′_{v} = K_{v} ∩ g_{v}^{−1}K_{v}g_{v}; On the other hand, for the archimedian place v we have

Sol_{∞}(γ) = {x_{∞} ∈ L_{∞}(R)/K̃_{∞} | x_{∞}^{−1}γ_{L}x_{∞} ∈ K̃_{∞} } .

With these definitions we get

F ix(γ) = L(Q) \ (L(Q) · ∏_{v} Sol_{v}(γ)) .

The operation of δ ∈ L(Q) via x_{v} ↦ x_{v}δ gives δ · ∏ Sol_{v}(γ) = ∏_{v} Sol_{v}(δγδ^{−1}).

\underline{Remark}: It is shown in the appendix 2, that S_{∞}(γ) = L_{γ,∞}/K̃ ∩ L_{γ,∞} = L_{γ,∞}^{0}/L̃_{γ,∞}^{0} ∩ K̃ is the symmetric space attached to L_{γ}^{0}.

4) From the hypothesis K small we get the simpler expression

F ix(γ) = L_{γ}(Q) \ ∏_{v} Sol_{v}(γ) .

L(Q) equivalence within Sol(γ) give rise to x^{−1}n_{1}γx ∈ g^{−1}KK̃_{∞} and x^{−1}n_{2}δ^{−1}γδx ∈ g^{−1}KK̃_{∞} for suitable n_{1}, n_{2} ∈ N(A A_{f}) and δ ∈ P (Q). It implies x^{−1}nδ^{−1}γδγ^{−1}x ∈ KK̃_{∞} for some n ∈ N(A A). Because K is small, this implies δ_{L}^{−1}γ_{L}δ_{L}γ_{L}^{−1} ∈ Z_{L}(Q). On the other hand δ_{L}^{−1}γ_{L}δ_{L}γ_{L}^{−1} ∈ L_{der}(Q). But Z_{L}(Q) ∩ L_{der}(Q) is finite. Hence the commutator is a torsion element and therefore equal to 1, again by our assumption on K.

5) Consider the finite places, let Ω be the stabilizer of a special point in the Bruhat-Tits building. They always exist. Ω is a maximal compact subgroup of G(A A_{f}). We can assume K small enough, such that K ⊂ Ω without restriction of generality. Then we have the Iwasawa decomposition G(A A_{f}) = P (A A_{f}) · Ω (see [T], 3.3.2). For k ∈ Ω put

g_{k}^{−1} = kg^{−1}k^{−1} and K_{k} = kKk^{−1} and K′_{k} = kK′k^{−1}. Then Sol(γ) = ∏_{v} Sol_{v}(γ) is

Sol(γ) = ⋃_{k∈P (A A_{f})∩Ω\Ω/K′} Sol(γ, k)

Sol(γ, k) = ∏_{v} Sol_{v}(γ, k) ,

where Sol_{∞}(γ, k) = Sol_{∞}(γ) and Sol_{f}(γ, k) = ∏_{v≠∞} Sol_{v}(γ, k) and

Sol(γ, k) = {p ∈ N(A A_{f}) \ P (A A_{f})/(P (A A_{f}) ∩ Ω ∩ kK′k^{−1}) | p_{L}^{−1}γ_{L}p_{L} ∈ (g_{k}^{−1}K_{k} ∩ P)_{L}}
= {m ∈ L(A A_{f})/(P (A A_{f}) ∩ K′_{k})_{L} | m^{−1}γ_{L}m ∈ (g_{k}^{−1}K_{k} ∩ P)_{L}} .

With these notations we get

L_{γ}(Q) \ Sol(γ, k) = L_{γ}(Q) \ S(γ, k)/K_{A A} (k) , K_{A A} (k) = K̃_{∞}(P (A A_{f}) ∩ K′_{k})_{L} ,

where

S(γ, k) = {m ∈ L(A A) | m^{−1}γ_{L}m ∈ K̃_{∞}(g_{k}^{−1}K_{k} ∩ P (A A_{f}))_{L}} .

The topological connected component of L_{γ,∞} acts transitively on each of the topological connected components of S(γ, k)/K_{A A}. Let denote L_{γ}^{0} the algebraic connected component of L_{γ}, considered as a reductive algebraic group over Q. Choose a decomposition

S(γ, k) = ⨄_{ξ} L_{γ}^{0}(A A) · ξ · K_{A A} (k)

with representatives ξ ∈ L(A A). Representatives ξ = ∏_{v} ξ_{v} can be chosen as products of corresponding local representatives ξ_{v} for L_{γ}^{0}(Q_{v}) \ S_{v}(γ, k)/(K_{A A} )_{v}. According to [K], 7.1 and 8.2(c) for almost all v there is only one representative ξ_{v} = 1. There is also only one representative ξ_{∞} = 1 at the infinite place (see appendix 2). Hence the representatives are essentially finite adeles ξ = ξ_{f} giving also the corresponding decomposition over the finite adeles. In fact, the local finiteness of the number of representatives for a fixed place follows from the fact, that the map L(Q_{v}) → L_{γ}^{0}(Q_{v}) \ L(Q_{v}) is open. We conclude

S(γ, k)/K_{A A} (k) = ⨄_{ξ} L_{γ}^{0}(A A)/ξK_{A A} (k)ξ^{−1} ∩ L_{γ}^{0}(A A)

This decomposes S(γ, k)/K_{A A} (k) into finitely many orbits under L_{γ}^{0}(A A). The image of each orbit in the quotient modulo L_{γ}^{0}(Q) can be identified with the space X = L_{γ}^{0}(Q) \ L_{γ}^{0}(A A)/ξK_{A A} (k)ξ^{−1} ∩ L_{γ}^{0}(A A). By assumption K is small. Hence L_{γ}(Q) ∩ m(K̃_{∞}(P (A A_{f}) ∩ K′_{k})_{L})m^{−1} is contained in the center Z_{L}(Q) of L. But Z_{L} ⊂ L_{γ}^{0}. This implies, that L_{γ}(Q)/L_{γ}^{0}(Q) acts freely on L_{γ}^{0}(Q) \ Sol(γ, k). Therefore

χ(L_{γ}(Q) \ Sol(γ, k)) = [L_{γ}(Q) : L_{γ}^{0}(Q)]^{−1} ∑_{ξ} χ(L_{γ}^{0}(Q) \ L_{γ}^{0}(A A)/ξK_{A A} (k)ξ^{−1} ∩ L_{γ}^{0}(A A))

Observe γ is R-elliptic, hence Q-elliptic. For the ambiguity of this notion see [K], p.392. We will show, that in our situation this ambiguity does not cause any trouble, because the Euler characteristic of the corresponding summands in the trace formula vanishes unless both notions agree. Consider the group L or better L/Z_{L}. The center Z(Z_{γ}^{0}) is Q-anisotropic modulo Z_{L}. If the quotient were not anisotropic over R, the corresponding global quotient space X is a nontrivial torus fibration, whose Euler characteristic therefore vanishes. Similarily for locally symmetric arithmetic quotients of semisimple groups the Euler characteristics vanishes unless the R-rank of the maximal compact subgroup equals the R-rank of the group. Hence we can assume, that (L_{γ}^{0}/Z_{L})(R) contains an R-anisotropic torus of maximal rank. Otherwise the corresponding summand does not contribute to the trace formula. By assumption A1 the center Z_{G} and therefore also the center Z_{L} is R-anisotropic modulo its Q-split torus. The same therefore holds for the center of L_{γ}^{0}.

The group K is small. Therefore L_{γ}(Q) ∩ m(K̃_{∞}(P (A A_{f}) ∩ K′_{k})_{L})m^{−1} is contained in the center of L_{γ}. But this is a discrete and (by assumption A1 on Z_{G} essentially) compact, hence finite group and consists only of torsion. By our assumption K is small, hence the intersection is trivial. Therefore there is a constant χ(L_{γ}^{0}) = χ(L_{γ}^{0}, dg_{f}) (depending on a choice of Haar measure on L_{γ}^{0}(A A_{f})), such that

χ(L_{γ}^{0}(Q) \ L_{γ}^{0}(A A)/ξK_{A A} (k)ξ^{−1} ∩ L_{γ}^{0}) = χ(L_{γ}^{0}, dg_{f})vol_{dg_{f}} (ξK_{A A_{f}} (k)ξ^{−1} ∩ L_{γ}^{0}(A A_{f}))^{−1} .

This is reduced to the following cases: the semisimple case, the case of a split torus, the case of Q-anisotropic tori, which are also R-anisotropic. See also [A], (6.3) and [A], p. 283 and [Ha], [Se].

6) As explained in section 2, the Lefschetz number therefore becomes the sum

L(f, λ) = ∑_{P} ∑_{γ∈L(Q)/∼} Ψ(γ_{L}, λ) ∑_{k∈P (A A_{f})∩Ω\Ω/K′} r(γ, k)χ(L_{γ}(Q) \ Sol(γ, k))

taken over all standard Q-parabolic subgroups P of G.

Now we use r(γ, k) = |γ|_{∞}^{−2ρ_{P}} r(γ^{−1}, k) from lemma 1 and 2. Put χ(γ, k) = χ(L_{γ}(Q)\Sol(γ, k).

Then it remains to understand the sum

∑_{k∈P (A A_{f})∩Ω\Ω/K′} r(γ^{−1}, k)χ(γ, k) .

Using lemma 3 we will show, that it is equal to

∑_{k∈Ω/K′} vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k}) · vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L})χ(γ, k) .

We take this for granted at the moment. Then we get the following formula for the Lefschetz number

L(f, λ) = ∑_{P} ∑_{γ∈L(Q)/∼} Ψ(γ, λ)|γ|_{∞}^{−2ρ_{P}} L_{P,γ}(f)

L_{P,γ}(f) = ∑_{k∈Ω/K′} vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k}) · vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L})[L_{γ}(Q) : L_{γ}^{0}(Q)]^{−1}×

χ(L_{γ}^{0}(Q) \ S(γ, k)/K̃_{∞}(P (A A_{f}) ∩ K′_{k})_{L})

Using the coset decomposition of 5), the Euler characteristics are given by

∑_{ξ} χ(L_{γ}^{0}, dg_{f})vol_{dg_{f}} (ξK_{A A} (k)ξ^{−1} ∩ L_{γ}^{0}(A A_{f}))^{−1} .

They depend on the choice of a Haar measure dg_{f} on the groups L_{γ}^{0}(A A) of points over the finite adeles. To fix it choose a Tamagawa measure on L_{γ}^{0}(A A), if γ is not central, and let dg_{f} be its finite part. It remains to simplify this expression for L_{P,γ} in the next section (see the result in lemma 4). Before we do that, we have to complete the proof of the argument using lemma 3, mentioned above.

\underline{Computation of}:

∑_{k∈Ω/K′} vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k}) · vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L})χ(γ, k) .

We divide by P (A A_{f}) ∩ Ω and get

∑_{k}[P (A A_{f}) ∩ Ω : P (A A_{f}) ∩ K′_{k}] · vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k}) · vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L}) · χ(γ, k) ,

where the sum is over all k ∈ P (A A_{f}) ∩ Ω \ Ω/K′. But

[P (A A_{f}) ∩ Ω : P (A A_{f}) ∩ K′_{k}] vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k})vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L}) =

vol_{P (A A_{f})}(P (A A_{f}) ∩ Ω)vol_{P (A A_{f})}(P (A A_{f}) ∩ K′_{k})^{−1}×
vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k})vol_{L(A A_{f})}((K′_{k} ∩ P (A A_{f}))_{L})
= vol_{N(A A_{f})}(N(A A_{f}) ∩ K′_{k})^{−1}vol_{N(A A_{f})}(N(A A_{f}) ∩ K_{k})[N(A A_{f}) ∩ K_{k} : N(A A_{f}) ∩ K′_{k}] ,

where measures are normalized by vol_{P (A A_{f})}(P (A A_{f}) ∩ Ω) = 1. On the other hand the last

index is [N(A A_{f}) ∩ kKk^{−1} : N(A A_{f}) ∩ kK′k^{−1}] or [pN(A A_{f})p^{−1} ∩ pkK(pk)^{−1} : pN(A A_{f})p^{−1} ∩ pkK′(pk)^{−1}]. It is equal to [N(A A_{f}) ∩ x_{f}Kx_{f}^{−1} : N(A A_{f}) ∩ x_{f}K′x_{f}^{−1}] = [K_{N}(h) : K′_{N}(h)]. This

was computed in lemma 3 as r(γ^{−1}) = r(γ^{−1}, k). This verifies the claim made above.

7.Computation of an orbital integral:

Our intention is, to rewrite certain terms of the Lefschetz numbers in the form of orbital integrals O_{γ}^{L}(\overline{f}^{(P)}). This is done in step 1-3. The result is formulated in step 4.

1) Assume measures are normalized by vol_{G}(Ω) = 1. Remember K′ = K ∩ g^{−1}Kg ang g ∈ G(A A_{f}) is fixed. Then the characteristic function 1_{g^{−1}K}(y) of the set g^{−1}K is K′ biinvariant. Furthermore k^{−1}xk ∈ g^{−1}K means x ∈ kg^{−1}k^{−1}kKk^{−1}. Therefore

∫_{Ω} 1_{g^{−1}K}(k^{−1}xk)dk = [Ω : K′]^{−1} ∑_{k∈Ω/K′} 1_{g^{−1}K}(k^{−1}xk) = vol_{Ω}(K′) ∑_{k∈Ω/K′} 1_{g_{k}^{−1}K_{k}}(x) .

For all x ∈ G(A A_{f}) we have

∫_{K} 1_{g^{−1}K}(k^{−1}xk)dk = ∫_{K} 1_{g^{−1}K}(k^{−1}x)dk = vol_{Ω}(g^{−1}Kg ∩ K)1_{Kg^{−1}K}(x) = vol_{Ω}(K′)1_{Kg^{−1}K}(x) .

Using ∫_{Ω} = vol_{Ω}(K)^{−1} ∫_{Ω} ∫_{K} we get

∫_{Ω} 1_{g^{−1}K}(k^{−1}xk)dk = vol_{Ω}(K′) ∫_{Ω} vol_{Ω}(K)^{−1}1_{Kg^{−1}K}(k^{−1}xk)dk .

This gives a formula for the Ω-average

\underline{Definition 1}: \overline{f}(x) = ∫_{Ω} f(k^{−1}xk)dk , x ∈ G(A A_{f})

where we put (see also appendix 1)

\underline{Definition 2}: f(x) = vol_{Ω}(K)^{−1}1_{Kg^{−1}K}(x).

With these definitions we got: \overline{f}(x) = ∑_{k∈Ω/K′} 1_{g_{k}^{−1}K_{k}}(x).

2) Let P = LN be a parabolic subgroup. Assume x ∈ P (A A_{f}). Then by step 1)

∫_{N(A A_{f})} \overline{f}(xn)dn = ∑_{k∈Ω/K′} ∫_{N(A A_{f})} 1_{g_{k}^{−1}K_{k}}(xn)dn .

If x_{L} is not in (g_{k}^{−1}K_{k} ∩ P (A A_{f}))_{L}, the corresponding integral on the right side vanishes. Otherwise xn_{0} = g_{k}^{−1}k_{0}. Then the corresponding integral is ∫_{N(A A_{f})} 1_{g_{k}^{−1}K_{k}}(g_{k}^{−1}k_{0}n)dn =

vol(N(A A_{f}) ∩ K_{k}). Hence for x ∈ L(A A_{f}) we get

ϕ(x) := ∫_{N(A A_{f})} \overline{f}(xn)dn = ∑_{k∈Ω/K′} vol(N(A A_{f}) ∩ K_{k})1_{(g_{k}^{−1}K_{k}∩P (A A_{f}))_{L}}(x) .

3) Next consider the orbital integral of this function ϕ on the group L(A A_{f})

O_{γ}^{L}(ϕ) = ∫_{L_{γ}^{0}(A A_{f})\L(A A_{f})} ϕ(m^{−1}γm)dm .

We will see, that it is related to some part of the local Lefschetz contribution L_{P,γ}(f). We defined certain sets Sol_{f}(γ, k) in section 6.5. From their definition and the definition of the function ϕ we immediately get

O_{γ}^{L}(ϕ) = ∑_{k∈Ω/K′} vol(N(A A_{f}) ∩ K_{k})

× ∫_{L_{γ}^{0}(A A_{f})\L(A A_{f})} char{m | m^{−1}γ_{L}m ∈ (g_{k}^{−1}K_{k} ∩ P (A A_{f}))_{L}}(m)dm ,

or equivalently

= ∑_{k∈Ω/K′} vol(N(A A_{f}) ∩ K_{k})vol((K′_{k} ∩ P (A A_{f}))_{L}) ∫_{L_{γ}^{0}(A A_{f})\L(A A_{f})/(K′_{k}∩P (A A_{f}))_{L}} 1_{Sol_{f} (γ,k)}(m)dm .

Now decompose Sol_{f}(γ, k) ⊂ L(A A) into ξ-cosets L_{γ}^{0}(A A_{f}) · ξ_{f}, using the notation of section 6.5. This gives

∫_{L_{γ}^{0}(A A_{f})\L(A A_{f})/(K′_{k}∩P (A A_{f}))_{L}} 1_{S(γ,k)}(m)dm

= ∑_{ξ_{f}} vol_{L_{γ}^{0}(A A_{f})}(ξ(K′ ∩ P (A A_{f}))_{L}ξ^{−1} ∩ L_{γ}^{0}(A A_{f}))^{−1} .

4) Let us put things together. The functions

\overline{f}(x) = ∫_{Ω} f(k^{−1}xk)dk , f(x) = vol_{Ω}(K)^{−1}1_{Kg^{−1}K}(x)

from our definitions are K biinvariant on G(A A_{f}). Put

\overline{f}^{(P)}(m) = |m|_{f}^{ρ_{P}} ∫_{N(A A_{f})} \overline{f}(mn)dn , m ∈ L(A A_{f})

O_{γ}^{L}(\overline{f}^{(P)}) = ∫_{L_{γ}^{0}(A A_{f})\L^{0}(A A_{f})} \overline{f}^{(P)}(m^{−1}γm)dm .

Assume that the integrals are normalized by the conventions vol_{P (A A_{f})}(Ω ∩ P (A A_{f})) = 1, vol(Ω) = 1 and vol_{N(A A_{f})}(Ω ∩ N(A A_{f})) = 1. See also [C], p.144. The computation above shows, that O_{γ}^{L}(\overline{f}^{(P)}) covers all terms in the formula for the local Lefschetz number except [L_{γ}(Q) : L_{γ}(Q)^{0}]^{−1}χ(L_{γ}^{0}, dg_{f}). We should take into account, that |γ|_{∞}^{−ρ_{P}} = |m^{−1}γm|_{f} is a factor, which we borrowed in the definition of \overline{f}^{(P)}. Now insert the result of the previous computation into the topological trace formula L(f, E_{λ}) = ∑_{P,γ} L_{P,γ}(f). As any K-biinvariant function f is a linear combination of functions as above, we get the

\underline{Lemma 4}: Assume G satisfies condition A1. Then for allowable Hecke operators f (see appendix 1) we have

L(f, E_{λ}) = ∑_{P} ∑_{γ∈L(Q)/∼}[L_{γ}(Q) : L_{γ}(Q)^{0}]^{−1} · χ(L_{γ}^{0})O_{γ}^{L}(\overline{f}^{(P)}) · |γ|_{∞}^{−ρ_{P}} Ψ(γ_{L}, λ) .

The sum extends over all P = LN containing the fixed minimal parabolic P_{0} and the conjugacy classes of semisimple, R-elliptic elements γ in L(Q) which are P-contractive.

\underline{Remark 1}: Because of −ρ_{G} + ρ_{P} = ρ_{L}, the factor |γ|_{∞}^{−ρ_{P}} Ψ(γ_{L}, λ) is equal to

∑_{w∈W^{P}} (−1)^{l(w)}ψ_{w(λ+ρ_{G})−ρ_{L}}(γ_{L}^{−1}) .

\underline{Remark 2:} The factor χ(L_{γ}^{0}, dg_{f})O_{γ}^{L}(.) does not depend on the choice of the fixed Haar measure dg_{f} on L_{γ}^{0}(A A_{f}).

\underline{Remark 3:} Suppose P = P_{θ}. Then γ is P-contractive, if |α(γ)|_{∞} > 1 or equivalently |α(γ)|_{f} < 1 holds for all simple roots α ∉ θ.

8.The Satake transform

Consider a nonarchimedian place v. Assume P_{v} = L_{v}N_{v} is a parabolic subgroup of G_{v}. Let σ_{v} be an admissable representation of a reductive groups L_{v} over the local field corresponding to v. Any function f_{v} in the Hecke algebra of locally constant functions with compact support on G_{v} acts on the unitarily normalized induced representation π_{v} = Ind_{P_{v}}^{G_{v}}(σ_{v}) by convolution. For a suitable choice of measures we have the adjunction formula

tr(f_{v}, Ind_{P_{v}}^{G_{v}}(σ_{v})) = tr(\overline{f}_{v}^{(P)}, σ_{v})

This formula is well known. See e.g. [K], cuspidal geometry of p-adic groups, lemma 1. The slightly different definition involving f_{v}^{∗} in loc, cit. has no effect.

Now assume P_{v} to be a \underline{minimal} parabolic subgroup of G_{v} and assume G_{v} to be quasisplit and split over a finite unramified extension field. Let A_{v} be a maximal split central torus of P_{v}. For elements f_{v} in the spherical Hecke algebra of G_{v} the Satake transform ([C],(19)) is defined by \overline{f}_{v}^{(P)} and gives a function on L_{v}(Q_{v})/L_{v}(Q_{v}) ∩ K_{v} = Λ. The group Λ is (commensurable with) the group of cocharacters X_{∗}(A_{v}) of A_{v} ([C] p.135). The Hecke operator f_{v} will be called \underline{regular} or of regular support for its Satake transform, if the image of the support of \overline{f}_{v}^{(P)} with respect to the natural map to X_{∗}(A_{v}) ⊗ R is disjoint from the walls of the root system.

\underline{Lemma 5}: Fix some nonempty open cone C in the positive Weyl chamber of the root system attached to P_{v}. Let π_{v} = Ind_{P_{v}}^{G_{v}}(σ_{v}) be an induced representation with a spherical constituent π_{v}^{0}. Then there exist spherical Hecke operators f_{v} with the following two properties:

1) tr(f_{v}, π_{v}) = tr(f_{v}, π_{v}^{0}) = 1.

2) The support of the Satake transform \overline{f}^{(P)} of f is contained in the Weyl group

orbit ⋃_{w∈W} w(x_{0} + C) of x_{0} + C.

x_{0} ∈ C can be chosen arbitrarily large.

\underline{Proof}: The Satake transform maps the spherical Hecke algebra of G_{v} onto the W invariant elements of the spherical Hecke algebra of L_{v}. Suppose the statement were not true. Then there exist finitely many different characters χ_{i}, i = 1, .., r (in the orbit of σ on the group Λ), such that

∑_{i} n_{i}χ_{i}(x) = 0 , (n_{1}, .., n_{r}) ≠ 0

holds for all x ∈ L_{v} in the preimage of x_{0} + C ⊂ Λ of the interior of the positive Weyl chamber. We will show, that this is impossible. First assume x_{0} = 0. Use that x, y ∈ C implies x + y ∈ C. To prove linear independence of characters on C, we can argue by lowering the lenght r of any nontrivial character relation on C using subtractions. This leads to a contradiction. However each subtraction step requires existance of elements y ∈ C such that χ_{r}(y) − χ_{1}(y) ≠ 0. Remember, any nontrivial character χ = χ_{r}/χ_{1} of Λ is different from 1 on Λ^{+}. Otherwise the kernel contains C, as any element of Λ can be written as a difference of two elements in C ∩ Λ (C is an open cone). This gives the contradiction. For x_{0} ≠ 0 the same argument works. Simply reduce to the case x_{0} = 0 by changing the coefficients n_{i} into n_{i}χ_{i}(x_{0}). This proves lemma 5.

\underline{Relative support}: Let f_{v} be a Hecke operator on G_{v} and P_{v} be a minimal parabolic subgroup. Consider the support of O_{γ}^{L}(\overline{f}_{v}^{(Q)}) as a function of γ ∈ L_{v}(Q_{v}) for intermediate parabolic group P_{v} ⊂ Q_{v} ⊂ G_{v}. Suppose Q_{v} = L_{v}N_{v}. Let A_{Q_{v}} ⊂ L_{v} be a maximal split central subtorus of L_{v}. Then the support will be considered as a subset of X_{L} = X^{∗}(A_{Q_{v}}) ⊗ R as follows. We have a surjective map

L_{v}(Q_{v}) → (L_{v})^{ab}(Q_{v}) ,

as L_{der} is connected and simply connected (assumption A1). Let the index ^{0} denote maximal compact subgroups ^{0}T in the case of tori T. Then one has a surjection

L_{v}^{ab}(Q_{v}) → L_{v}^{ab}(Q_{v})/^{0}L_{v}^{ab}(Q_{v}) .

The kernel of the natural map A_{Q_{v}} → L_{v}^{ab} is contained in ^{0}A_{Q_{v}}(Q_{v}). Therefore the group A_{Q_{v}}(Q_{v})/^{0}A_{Q_{v}}(Q_{v}) injects into L_{v}^{ab}(Q_{v})/^{0}L_{v}^{ab}(Q_{v}) as a subgroup of finite index. It can be identified with X_{∗}(A_{Q_{v}}). Namely each a in this cosets can be evaluated on a character χ to give ord(χ(a)) ∈ ZZ. Thus L_{v}^{ab}(Q_{v})/^{0}L_{v}^{ab}(Q_{v}) can be viewed as a lattice in X_{L} := X^{∗}(A_{Q_{v}}) ⊗ R. This allows to define the composite map

π_{L} : L_{v}(Q_{v}) → L_{v}^{ab}(Q_{v})/^{0}L_{v}^{ab}(Q_{v}) → X_{L} .

The map π_{L} : L(Q_{v}) → X_{L} has the following obvious properties: It is invariant under conjugation and under left and right multiplication. For an arbitrary function χ : X_{L} → R, the multiplication by χ determines an endomorphism of the Hecke algebra of L_{v}, which preserves the subspace of spherical Hecke operators on L_{v}. Note the trivial property

O_{γ}^{L}(gχ) = χ(π_{L}(γ))O_{γ}^{L}(g) .

The image of the regular, semisimple support of the function O_{γ}^{L}(\overline{f}_{v}^{(Q)}) on γ ∈ L_{v}(Q_{v}) in X_{L} under the map π_{L} is called the \underline{relative} support of f_{v} with respect to Q_{v}. It is a finite subset of X_{L}. Let χ_{x} be the Dirac function for x ∈ X_{L}. Then the following statements are equivalent

i) x ∈ X_{L} is in the relative support of f_{v}.

ii) O_{γ}^{L}(\overline{f}^{(Q)}χ_{x}) ≠ 0 for some semisimple, regular γ ∈ L_{v}(Q_{v})

From now let us assume, that f_{v} is a spherical Hecke operator on G_{v}. Then \overline{f}_{v}^{(Q)} is spherical on L_{v} with respect to K_{L} = K ∩L. The description of spherical Hecke operators in terms of the Satake isomorphism shows that the algebra of spherical Hecke operators is mapped isomorphically onto the space of W invariant spherical Hecke operators on the minimal Levi component, by the map f_{v} ↦ S(f_{v}) = \overline{f}_{v}^{(P_{v})}. (See [C], p. 146 ff). Property ii) implies that \overline{f}^{(Q)} is not zero. Therefore ii) is equivalent to

iii) O_{γ}^{L}(\overline{f}^{(Q)}χ_{x}) ≠ 0 for some semisimple, regular γ contained in the maximal split torus A_{v} of L_{v} or

iv) S(f_{v})χ_{x} ≠ 0.

This descibes the relative support for Q_{v} in terms of the support of the Satake transform, i.e. in terms of the relative support in in X = X_{∗}(A_{v}) ⊗ R. We have the natural map A_{v}(Q_{v}) → L_{v}(Q_{v}) → X_{L}, which factors over the quotient A_{v}(Q_{v})/^{0}A_{v}(Q_{v}) and induces a canonical R-linear map

pr_{L} : X → X_{L} .

This gives another version of iv)

v) x ∈ X_{L} is in the pr_{L}-image of the relative support in X of the Satake transform of f_{v}.

The embedding A_{Q_{v}} ↪ A_{v} induces canonical embeddings

X_{L} = X_{∗}(A_{Q_{v}}) ⊗ R ↪ X = X_{∗}(A_{v}) ⊗ R .

We have to consider the projections

pr_{L} : X = X_{∗}(A_{v}) ⊗ R → X_{L} = X_{∗}(A_{Q_{v}}) ⊗ R .

pr_{L} restricts to the identity map on the subspace X_{L}. Let A′_{Q_{v}} be the split torus L_{der} ∩A_{v}. Then pr_{L} annilitates X_{∗}(A′_{Q_{v}}) ⊗ R. These two properties determine pr_{L}.

We give the well known reformulation in terms of the Killing form. Let α_{i} denote the simple Q_{v}-roots, let ⟨ , ⟩ denote the Killing form and let β_{j} denote the dual basis ⟨α_{i}, β_{j}⟩ = δ_{ij}. Use the Killing form to identify X_{∗}(A_{v}) ⊗ R with its dual X^{∗}(A_{v}) ⊗ R. Then X_{∗}(A_{Q_{v}})⊗ R is given in X_{∗}(A_{v})⊗ R by the equations, say ⟨ ., α_{i}⟩ = 0, i ∈ F. The X_{∗}(A_{P,v})⊗ R splits into the orthogonal direct sum of the two subspaces X_{∗}(A_{Q_{v}}) ⊗ R = ∑_{j∉F} Rβ_{j} and ∑_{i∈F} Rα_{i}. The map pr_{L} is therefore the orthogonal projection

pr_{L}(∑_{j∉F} x_{j}β_{j} + ∑_{i∈F} y_{i}α_{i}) = ∑_{j∉F} x_{j}β_{j} .

\underline{Subdivision of the chambers}:

Suppose Q_{v} = L_{v}N_{v}. Then a regular semisimple element γ ∈ L_{v}(Q_{v}) is Q_{v}-contractive, i.e. satisfies |α(γ)|_{v} < 1 for all roots α of A_{Q_{v}} on N_{Q_{v}}, if and only if its image in the relative support is contained in the positive Weyl chamber of X_{L}. Remember, the positive Weyl chamber is the set of all x ∈ X_{∗}(A_{Q_{v}}) ⊗ R, such that α(x) > 0 holds for all nonvanishing restrictions of simple positive roots α ∈ Δ ⊂ X^{∗}(A_{v}), i.e. for the α ∉ θ, if Q = P_{θ}.

The complement of the singular hyperplanes in the real vector space X_{L}, defined by the equations α(x) = 0, decomposes into open connected cones. One of them is the positive Weyl chamber of X_{L}.

Consider the orthogonal projections pr_{L} : X → X_{L}. The preimages of all singular hyperplanes in X_{L}, for the finitely many standard parabolic groups, define finitely many hyperplanes in X. Symmetrise them under the Weyl group W. This gives a finite number of hyperplanes of X. They dissect the positive Weyl chamber into finitely many open connected subcones after removing this set of hyperplanes. This defines a finite subdivision of the Weyl chambers of X.

\underline{Example}: Suppose G_{v} = Sl(3, Q_{v}). Let α_{1}, α_{2} be a root basis of type A_{2}. Then the process described above divides the positive Weyl chamber in two subcones. The dissection cuts the positive Weyl chamber in the direction of the root α_{1} + α_{2}.

\underline{Now turn to a global situation}: Consider a finite set of inequivalent irreducible automorphic representations on G(A^{S}), S some finite fixed set of bad places containing the archimedian places. Fix one of them, say π^{S} = ⊗_{v∉S}π_{v}. A locally constant function f_{π^{S}} with compact support on G(A A^{S}) is called a π^{S}-projector, if its trace on all the finitely many representations vanishes except for π^{S}, where the trace should be one. Suppose we have given such a f_{π^{S}}. By enlarging S we may assume, that all the finitely many given representations are spherical outside S. Choose a large auxiliary place v outside the fixed finite set S. We may assume that (f_{π^{S}})_{v} = 1_{K_{v}}.

Construct f_{v} as in lemma 5, with respect to the fixed local spherical representation π_{v}^{S} at the auxiliary place v. Then, if we replace f_{π^{S}} by f_{π^{S}}f_{v}, this still gives a π^{S}-projector.

Let it again denote f_{π^{S}}, by abuse of notation. This v-modified projector will be called a good π^{S}-projector, if it behaves well in the following sense:

\underline{Good π^{S}-projectors}:

Consider g ∈ P (A A_{f}). Then x^{−1}gx ∈ ω for a compact subset ω ⊂ L(∏_{w≠v} A A_{v}), implies, using Iwasawa decomposition x = pk, that the image of g in L^{ab}(A A_{f}) is contained in a fixed compact subset. Therefore the product of orbital integrals

∏_{w≠v finite} O_{g}^{L_{w}}(\overline{f}_{π^{S},w}^{(Q)})

has compact ”relative” support in L^{ab}(A A_{f}^{v}) (image of the support of the function on ∏_{w≠v finite} L_{w}). Let it denote supp(f_{π^{S}}^{v}). Let us turn to the modification place v. Remember, the modifying function f_{v} was constructed as in lemma 5. The cone C will be specified to be one of the subcones of the positive Weyl chamber, constructed above. Suppose x_{0} is a sufficantly large point in C. Then the relative support of f_{v} in X is in the orbit of x_{0} + C under the Weyl group. Suitably increase x_{0}. This forces the relative support to be arbitrarily far away from the walls. For a suitable choice of x_{0}, depending on the given supp(f_{π}^{v}), we get: An element γ in the support of O_{γ}^{L}(\overline{f_{π}f_{v}}^{(P)}) is P-contractive iff the local component γ_{v} is P_{v}-contractive at the specified place v of modification. Namely |α(γ)|_{∞} > 1 for a Q-root α holds iff |α(γ)|_{v} ∏_{w≠∞,v} |α(γ)|_{w} < 1. The term in the product is bounded from above and below depending on ω. On the other hand max(|α(γ)|_{v}, |α(γ)|_{v}^{−1}) > const(x_{0}) >> 1 for all γ in the support. This property can be achieved for all the finitely many standard parabolic subgroups L of G. In this case we call the modified projector f_{π} a good π-projector. It is clear, that good π-projector always exist.

In the special case, where the given set of representations is empty, the corresponding functions may simply be called \underline{good} Hecke operators.

\underline{Remark}: It is clear, that instead of one modification place v one can also use a finite set of modifying places. This is important for applications, where one considers Frobenius places within the set of modifying places.

9.Elliptic traces

Let us assume, that the Levi subgroups of G satisfy condition A1. Part of the condition A1 will be automatically inherited from G ([SS], thm. 5.8). Suppose that all \overline{f}^{(Q)} are allowable Hecke operators on the Levi subgroups.

Define elliptic ”traces”

T_{ell}^{G}(f, τ) = ∑_{γ∈G(Q)/∼}[G_{γ}(Q) : G_{γ}^{0}(Q)]^{−1}χ(G_{γ}^{0})O_{γ}^{G}(f) · tr(τ(γ^{−1}))

for finite dimensional rational representations τ of G(Q). The sum extends over the conjugacy classes of semisimple, R-elliptic elements in G(Q). The orbital integrals in this sum are taken with respect to the group of finite adeles. Similar define elliptic traces for the Levi subgroups.

Let χ_{P_{v}}^{G} be the characteristic function of the positive Weyl chamber of X_{L} = X_{∗}(A_{P}) ⊗ R, lifted to a function on L(Q_{v}). Suppose P = LN. Via the projection L(A A_{f}) → L(Q_{v}) we view it as a function on L(A A_{f}). Using the results of the last section, we obtain from lemma 4

\underline{Lemma 6}: Let f be a good allowable Hecke operator, modified at v. Then

L(f, E_{λ}) = ∑_{P_{0}⊂P ⊂G} T_{ell}^{P}(\overline{f}^{(P)}χ_{P_{v}}^{G}, λ)

where T_{ell}^{P}(h, λ) = ∑_{w∈W^{P}} (−1)^{l(w)}T_{ell}^{L}(h, w(λ + ρ_{G}) − ρ_{L}) and where P = LN. For simplicity

we did not distinguish between representations and their highest weights.

Let χ̂_{P_{v}}^{G} be the characteristic function of the obstuse Weyl chamber in X_{L}, considered as a function on L_{v}.

\underline{Lemma 7}: Let the situation be as in lemma 6. Then

T_{ell}^{G}(f, λ) = ∑_{P_{0}⊂Q⊂G}(−1)^{rang(Q)−rang(G)}L^{Q}(\overline{f}^{(Q)}χ̂_{Q_{v}}, λ)

where

L^{Q}(h, λ) = ∑_{w∈W^{Q}} L(h, E_{w(λ+ρ_{G})−ρ_{L}}) .

The latter are the Lefschetz numbers attached to the symmetric space of the Levi subgroup of the standard parabolic subgroup Q.

Lemma 6 and 7 were expected by Harder ([H], p. 144-145).

\underline{Proof}: Lemma 6 applied to the Levi subgroups L of Q = LN gives

L^{Q}(\overline{f}^{(Q)}χ̂_{Q_{v}}, λ)

= ∑_{P′=L′N′⊂L} ∑_{w∈W^{Q}}(−1)^{l(w)}T_{ell}^{P′}(\overline{(\overline{f}^{(Q)}χ̂_{Q_{v}}^{G})}^{(P′)}χ_{P′_{v}}^{L}, w(λ + ρ_{G}) − ρ_{L})

= ∑_{P′=L′N′⊂L} ∑_{w′∈W^{P′}} ∑_{w∈W^{Q}}(−1)^{l(w′)+l(w)}T_{ell}^{L′}(\overline{(\overline{f}^{(Q)}χ̂_{Q_{v}}^{G})}^{(P′)}χ_{P′_{v}}^{L}, w′w(λ + ρ_{G}) − ρ_{L′}) .

Here P ⊂ Q induces the parabolic group P′ = P ∩ Q in the Levi component L of Q. The Levi components L′ of P′ and P coincide. The character sn(w) = (−1)^{l(w)} satiesfies sn(w′)sn(w) = sn(w′w). From the inductivity W^{P′}W^{Q} = W^{P} and the formula \overline{f}^{(P)}χ̂_{Q_{v}}^{G} =

\overline{(\overline{f}^{(Q)}χ̂_{Q_{v}}^{G})}^{(P′)}, the sum simplifies

L^{Q}(\overline{f}^{(Q)}χ̂_{Q_{v}}, λ) = ∑_{P =L′N′⊂Q} T_{ell}^{P}(\overline{f}^{(P)}χ̂_{Q_{v}}^{G}χ_{P′_{v}}^{L}, λ) .

Summing these formulas over the standard parabolic groups Q, with the additional (−1) factors, gives the desired result. The sum ∑_{P ⊂Q⊂G}(−1)^{rang(G)−rang(Q)}χ̂_{Q_{v}}^{G}χ_{L′_{v}}^{L}, , for fixed P, is zero or one δ_{P G}. This is a well known result of Arthur.

Remark: It is clear, that analogous formulas are valid for all allowable Hecke operator with the global cutoff functions χ_{P}^{G} on L(A A_{f}).

10. Automorphic representations:

We now fix the level group K of G(A A). The G(A A_{f}) module lim_{K′}H^{•}(S_{K}(G), E_{λ}) defines an admissable representation. Therefore only finitely many irreducible constituents π of fixed level (π^{K} ≠ 0) occur. The same is true for the finitely many Levi subgroups L, the induced level groups and the induced coefficient systems attached to the representations

E_{λ′} for λ′ = w(λ + ρ_{G}) − ρ_{L}.

Let us define the following admissable automorphic representation

Π(λ) = ⨁_{P_{0}⊂Q⊂G} ⨁_{w∈W^{Q}} ⨁_{i} Ind_{Q}^{G}(lim_{K_{L}} H^{i}(S_{K_{L}}(L), E_{w(λ+ρ_{G})−ρ_{L})}) ,

the halo of the λ-cohomology of S(G). The space Π(λ) is a superspace. The grading is induced by the sign convention, given by the sum of the number rank(G) − rank(Q), the length l(w) for w ∈ W^{P} and the degree i.

Lemma 7 shows, that the elliptic ”trace” is almost an actual trace on Π(λ).

\underline{Assumption}: From now on we assume, that all Levi subgroups L of G satisfy condition A1 of the appendix. Furthermore we assume that not only (f, K) for f_{S} defines an allowable Hecke operator, but that also for all P the functions \overline{f_{S}}^{(P)}(am) define allowable Hecke operators on the Levi subgroup.

Consider a finite set S of places containing the bad places. Fix a level group K = K_{S}K^{S}, where K^{S} is a product of special good maximal compact subgroups. Fix λ, an irreducible constituent representation π = π^{S} of Π(λ), considered as a representation of the spherical Hecke algebra of G(A A_{f}^{S}) (finite places away from S). There are only finitely many constituents in Π(λ) with nontrivial K-invariants. Fix a finite set S_{π^{S}} disjoint from S to construct a projector f_{π^{S}}. Consider a finite set T of modifying places disjoint from S and S_{π^{S}}. Construct a good projector by modification at places v ∈ T. A place will be called a level place, if it is nonarchimedian and K_{v} is not a special, good maximal compact subgroup of G_{v}.

\underline{Theorem}: Suppose π is cuspidal and not of CAP type (see [PS] or [S]). For finitely many given f_{S}, there exists a good π^{S}-projector f_{π^{S}} such that

trace_{s}(f_{S}, H^{•}(S_{K}(G), E_{λ})(π^{S})) = T_{ell}^{G}(f_{S}f_{π^{S}}, λ) .

\underline{Proof}: Observe

tr(f_{w}, Ind_{Q_{w}}^{G_{w}}(σ_{w})) = tr(\overline{f}_{w}^{(Q_{w})}, σ_{w}) .

If we apply this for all w ≠ v, w ∈ S_{π} and the global representations σ occuring in the cohomology of proper Levi subgroups (with level K and infinity types determined by λ fixed), this implies

L^{Q}(\overline{(f_{S}f_{π^{S}})}^{(Q)}χ̂_{Q_{v}}, λ) = 0

for all Q ≠ G. This follows from a theorem of J. Franke, which shows that the representations σ appearing in cohomology, are automorphic. There the trace on the induced representation vanishes by the definition of good π^{S}-projectors, because π^{S} was assumed to be not a CAP representation. This proves, that in lemma 7 only T_{ell}^{G}(\overline{f}, λ) remains. But then we can replace \overline{f} by f. This follows from O^{G}(\overline{f}) = O^{G}(f), because of vol(Ω) = 1. This proves the theorem.

\underline{Remark 1}: In the hermitian symmetric case one can proof the analogous statement for L^{2}-cohomology instead of ordinary cohomology. See [GMP] for the relevant changes to be made in the arguments above. They only involve a change in the definition of W^{P}, which now also depend on γ. We get from this the following formula for the L^{2}-Lefschetz number

∑_{P =LN} ∑_{w∈W^{P}} (−1)^{I(w)}T_{ell}^{L}(\overline{f}^{(P)}χ_{P}^{G}(w), w(λ + ρ_{G}) − ρ_{L}) ,

where χ_{P_{v}}^{G}(w) depends on w ∈ W^{P}. The (global) cutoff functions χ_{P} (w) is the following. It is the characteristic function of the set of all γ ∈ L(Q), which satisfy #I(γ) = #I(w), where the I(w) are finite sets defined in [GMP], p.474. The number #I(γ) defined in [GMP], p. 471 is the number of roots α of A_{P} in N_{P}, such that |α(γ)|_{f}^{−1} = |α(γ)|_{∞} ≤ 1. Let χ_{P,i} be the (global) characteristic function of the set of all γ, such that |α_{i}(γ)|_{∞} > 1. Then the characteristic function χ_{P} (w) can be written in the form ∑_{#I=#I(w)} ∏_{i∈I} χ_{P,i} ∏_{j∉I}(1−χ_{P,j}). It can therefore be expanded into a finite linear combination of functions ∏_{i∈K} χ_{P,i}. These finitely many global cutoff functions, which appear in the L^{2}-Lefschetz formula, can be replaced by their local analogues ∏_{i∈K} χ_{P_{v},i}, provided f = f_{S}f_{π^{S}} and f_{π^{S}} is chosen to be a good π^{S} projector modified at v. This implies

\underline{Corollary}: Suppose π is cuspidal and not of CAP type. For finitely many f_{S} there exists a good π^{S}-projector f_{π^{S}} such that the L^{2}-Lefschetz number is

trace_{s}(f_{S}, H_{(2)}^{•}(S_{K}(G), E_{λ})(π^{S})) = T_{ell}^{G}(f_{S}f_{π^{S}}, λ) .

Especially the supertraces of allowable Hecke operators f_{S} on the π^{S}-isotypic subspace of cohomology and L^{2}-cohomology agree.

\underline{Remark}: If we consider an additional modifying place p and modify a good projector f_{π^{S}} by multiplication with a local Hecke operators at p, whose Satake transform has support sufficantly regular and is away from the modifying places, then the same conclusions hold. See also [H]. This very important observation will be used in [W] for the ”Frobenius” prime p.

11.The discrete series case

This is the case considered in [A]. We will compare with these results. The definition of arithmetic quotients there is slightly different, but by assumption A1 on Z_{G} actually agrees.

First observe

tr(τ(γ^{−1})) = tr(τ^{∗}(γ))

for the contragredient representation. Next one has the following formula for the Euler numbers [A], p.281 and 282

χ(G, dg_{f}) = (−1)^{q(G)}d(G)vol(G(Q)Z_{G,∞}^{0} \ G(A A))vol(Z_{G,∞}^{0} \ \overline{G}(R))^{−1} .

Here d(G) is the number of discrete series representations and 2q(G) the reell dimension of the symmetric domain attached to G_{∞}. Finally \overline{G}(R) is the anisotropic form of G_{∞}. Put

f_{∞} = (−1)^{q(G)}d(G)^{−1} ∑_{π∈Π_{disc}(τ)} f_{π_{∞}} .

Here f_{π_{∞}} denotes pseudo coefficients or the discrete series representations π_{∞} with respect to some fixed Haar measure dg_{∞}. Let γ be a semisimple, R-elliptic element from G(Q). Then the orbital integral

O_{γ}^{G_{∞}}(f_{∞}) = ∫_{G_{γ,∞}^{0}\G_{∞}} f_{∞}(gγg^{−1})dg_{∞}

equals

(−1)^{q(G_{γ}^{0})}d(G_{γ}^{0})/d(G)vol(Z_{G_{γ,∞}}^{0} \ \overline{G}_{γ,∞}^{0})^{−1}tr(τ^{∗}(γ)) .

This follows from [A], thm 5.1

O_{γ}^{G_{∞}}(f_{∞}) = (−1)^{q(G_{γ}^{0})}d(G_{γ}^{0})vol(Z_{G_{γ,∞}}^{0} \ \overline{G}_{γ,∞}^{0})^{−1} ∑_{ρ} ϕ_{G}(γ, ρ)trρ^{∗}(f_{∞})

using the notations of [A] without further comment. The sum runs over (certain) irreducible representations ρ of \overline{G}(R). This is understood in the sense of [A], p.271 formula (4.3)

trρ^{∗}(f_{∞}) = (−1)^{q(G)}trπ^{∗}(f_{∞})

for any discrete series representation π ∈ Π_{disc}(ρ). Using the definition of pseudo coefficients we find trρ^{∗}(f_{∞}) = 0 for ρ^{∗} not isomorphic to τ and trρ^{∗}(f_{∞}) = d(G)^{−1} if ρ^{∗} is isomorphic to τ. The formula for the orbital integral therefore follows from ϕ_{G}(γ, τ) = trτ^{∗}(γ) according to [A], p.271 and lemma 4.2 loc.cit.

Using the formulas above we conclude

T_{ell}^{G}(f, τ) = d(G) ∑_{γ∈G(Q)/∼}^{′} τ(G_{γ}^{0})[G_{γ}(Q) : G_{γ}^{0}(Q)]^{−1}O_{γ}^{G(A A )}(ff_{∞}) .

The summation is over all semisimple, R-elliptic conjugacy classes of G(Q). Here τ(G_{γ}^{0}) is the Tamagawa number vol(G_{γ}^{0}Z_{G_{γ,∞}^{0}} \ G_{γ}^{0}(A A)). Furthermore the measure dg_{∞} has to be chosen, such that dg_{∞}dg_{f} is the Tamagawa measure on G(A A).

\underline{Proof}: Suppose γ is not a central element. Then

χ(G_{γ}^{0})trτ(γ^{−1})O_{γ}^{G}(f)

= (−1)^{q(G_{γ}^{0})}d(G_{γ}^{0})vol(\overline{G}_{γ}^{0}(R)/Z_{G_{γ,∞}^{0}}^{0})^{−1}τ(G_{γ}^{0})trτ^{∗}(γ)O_{γ}^{G}(f)
= d(G)O_{γ}^{G_{∞}}(f_{∞})O_{γ}^{G}(f)τ(G_{γ}^{0})
= d(G)τ(G_{γ}^{0})O_{γ}^{G(A A )}(ff_{∞}) .

If γ is central, then the formula of course only holds, if the measure dg_{∞}dg_{f} gives the same volume as a Tamagawa measure.

Finally one should remark, that under the preliminary hypothesis A1 of the appendix actually G_{γ} = G_{γ}^{0} holds for all the γ. With this reformulation of the elliptic trace we get

\underline{Corollary}: We get

T r(f_{S}, H^{•}(S_{K}(G), E_{λ})(π)) = d(G) ∑_{γ∈G(Q)/∼}^{′} τ(G_{γ}^{0})[G_{γ} : G_{γ}^{0}]^{−1}O_{γ}^{G(A A )}(f_{S}f_{π^{S}}f_{∞})

under the assumptions on π^{S} and f_{π^{S}} of the theorem of the last section and the remarks thereafter. The measures for the orbital integrals are assumed to be Tamagawa measures on G(A A) and G_{γ}^{0}(A A).

\underline{Remark:} The term O_{γ}^{G}(f_{S}f_{π^{S}}f_{∞}) is independent from the measures dg_{f} and dg_{∞} chosen, provided dg_{∞}dg_{f} is the Tamagawa measure on G(A A). This follows from the definition of f_{π^{S}} and f_{∞}. Hence we are free to normalize the measures dg_{f} and dg_{∞}, such that vol_{dg_{f}} (K) = 1. This is the convention of [Ko2].

\underline{Remark}: In the theory of Shimura varieties (see [Ko2]) one replaces S_{K}(G) = G(Q) \ G(A A)/K̃K by G(Q) \ G(A A)/Zentr(h)_{∞}K, where h is the underlying structure homomorphism. For small K this multiplies the trace by a factor [K̃ : Zentr(h)_{∞}].

Appendix 1

Let G be a reductive connected group over Q and G_{der} its derived group. Consider the map ν : G → G/G_{der} = G^{ab}. Let K ⊂ G(A A_{f}) be a compact subgroup and Γ = K ∩ G(Q). For g ∈ G(A A_{f}) put

K′ = K_{g} = g^{−1}Kg ∩ K ⊂ K .

Consider M = G(Q) \ G(A A), or some compactification, with continuous G(A A_{f}) action m ↦ mk^{−1}, k ∈ G(A A_{f}) together with the maps p(m) = m resp. p′(m) = mg^{−1}

p : M/K_{g} → M/K

p′ : M/K_{g} → M/K .

The map p (resp. p′) is equivariant with respect to the map q (resp. q′)

K_{g} → K

given by k ↦ k resp. k ↦ gkg^{−1}. Two points mK and m′K in M/K are related by the correspondence defined by p, p′ if there exists a point m′′K_{g} ∈ M/K_{g} such that

p(m′′K_{g}) = mK and p′(m′′K_{g}) = m′K in M/K.

This means, that there exist k, k′ ∈ K, k′′ ∈ K_{g} such that

mkk′′ = m′′ and m′′g^{−1} = m′(k′)^{−1}

or mkk′′g^{−1}k′ = m′. In other terms this implies m′ = mx^{−1} for some x ∈ KgK. If KgK = ⨄_{j} Kg_{j} this is equivalent to

m′K = mg_{j}^{−1}K for some j.

Namely suppose conversely, that m′K = mkg^{−1}K for some k ∈ K. Then put m′′ = mkK_{g}.

We then get p(m′′K_{g}) = mkK = mK and p′(m′′K_{g}) = mkg^{−1}K = m′K.

We have to relate this condition for the adelic correspondence relation to its classical analog.

For that let us consider the following conditions A1, A2, A3. They involve the reductive group G, a compact open subgroup K ⊂ G(A A_{f}) of the group of point in finite adeles and an element g ∈ G(A A_{f}).

\underline{Assumption A1} (on the group G): G_{der} is connected, simply connected and (G_{der})_{∞} has no compact almost simple factor. Furthermore G^{ab} divided by a maximal Q-split torus is R-anisotropic.

\underline{Assumption A2} (on the pair (K, g)): Let ν : G → G^{ab} = G/G_{der} be the natural map. Then k ∈ K using A2. Then γ^{−1}gk^{−1}g_{∞}^{−1} ∈ G_{der}(A A) = ker(ν). By the strong approximation theorem γ^{−1}gk^{−1}g_{∞}^{−1} = γ_{d}g_{d,∞}k_{d}, where k_{d} ∈ K ∩ G_{der}(A A_{f}). The claim follows from

ν(g) ∈ ν(K)ν(G(Q))ν(G(R)) in G^{ab}(A A).

\underline{Assumption A3} (on the pair (K, g)): ν(K) = ν(K_{g})ν(Γ) in G^{ab}(A A_{f}), where Γ = K ∩ G(Q).

\underline{Claim 1}: A1 and A2 imply g ∈ image(G(Q) → G(A A_{f})) · K.

\underline{Proof}: Consider g in G(A A_{f}). Then ν(g) = ν(γ)ν(g_{∞})ν(k) for γ ∈ G(Q), g_{∞} ∈ G(R) and

g = (γγ_{d})(g_{d,∞}g_{∞})(k_{d}k).

\underline{Corollary}: Under the assumptions A1 and A2 the cosets KgK and gK have the form KγK and γK for some γ ∈ G(Q).

Now consider γ ∈ G(Q) and the double coset

ΓγΓ = ⨄_{i} Γγ_{i} .

Then

KγΓ = ⨄_{i} Kγ_{i}

is again a disjoint union. Observe k_{1}γ_{1} = k_{2}γ_{2} implies k_{2}^{−1}k_{1} = γ_{2}γ_{1}^{−1} ∈ G(Q) ∩ K = Γ. Therefore Γγ_{1} = Γγ_{2}.

\underline{Claim 2}: Assume A1, then KγΓK_{d} = ⨄_{i} Kγ_{i}, where K_{d} = K ∩ G_{der}(A A_{f}).

\underline{Proof}: As already shown, the right side is KγΓ, therefore contained in the left. The opposite inclusion follows from passing to the closure K_{d} of Γ∩G_{der}(Q) (strong approximation in G_{der}). Observe, the right hand side is a closed set.

\underline{Claim 3}: Assume A1, A2 and A3, then KγK = ⨄_{i} Kγ_{i}.

\underline{Proof}: It remains to show KγK ⊂ KγΓK_{d}. For that it is enough to show γK ⊂ KγΓK_{d} or K ⊂ γ^{−1}KγΓK_{d}. We now use condition A3. If ν(K_{g}) = ν(K_{γ}), this implies that for every element k ∈ K one can find some k′ ∈ ΓK_{γ} ⊂ K such that ν(k′k) = 1. Then k′k ∈ K ∩ G_{der} = K_{d} and k ∈ (k′)^{−1}K_{d} ⊂ K_{γ}ΓK_{d} ⊂ γ^{−1}KγΓK_{d} proves the inclusion required above.

For the proof of claim 3 it remains to show ν(K_{g}) = ν(K_{γ}). Remind the definition

K_{γ} = γ^{−1}Kγ ∩ K. We conclude ν(K_{γ}) = ν(γK_{γ}γ^{−1}) = ν(K ∩ γKγ^{−1}) = ν(K ∩ gKg^{−1}) =

ν(gK_{g}g^{−1}) = ν(K_{g}). The third equality follows from gK = γK.

\underline{The algebra H_{S,K}}:

There is a large class of elements g, that satisfy condition A2 and A3. Assume K = ∏ K_{v}. Consider the following situation:

There are groups Ω_{v} and A_{v} in G(Q_{v}), such that K_{v} is normal in Ω_{v} and K_{v} ∩ A_{v} is normal in A_{v}. Furthermore ν(K_{v}) = ν(A_{v} ∩ K_{v}) and G(Q_{v}) = Ω_{v}A_{v}Ω_{v} holds.

\underline{Example}: G_{v} split, K_{v} a normal subgroup of a maximal compact group Ω_{v} with Cartan decomposition, A_{v} a maximal split torus, such that K_{v} = (N^{+} ∩ K_{v})(A_{v} ∩ K_{v})(N^{−} ∩ K_{v}). There exists a cofinal system of such groups K_{v} in Ω_{v}.

\underline{Remark}: A special case are the principal congruence subgroups in the case of the groups

Gl(n) or GSp(2g).

\underline{Example}: Ω_{v} = K_{v} a special good, such that ν(K_{v}) = ν(A_{v} ∩ K_{v}) for a maximal split torus A_{v} holds. Furthermore assume Λ = X_{∗}(A) for v ∉ S in the sense of [C], p. 135. Then we have the Cartan decomposition in the form, that G_{v} is a disjoint union of double cosets K_{v}aK_{v}, where a runs over a set of representatives of A_{v}/^{0}A_{v}. Here ^{0}A_{v} is the maximal compact subgroup of the split torus A_{v}. It follows, that ^{0}A_{v} = A_{v} ∩ K_{v}.

Assume there is a finite set S of primes, such that outside S the assumption above is satisfied. Then consider all elements g = ∏_{v} g_{v}, such that g_{v} = 1 holds for all v ∈ S. This class of elements defines a subgroup of G(A A_{f}). Condition A3 is easily checked using the Cartan decomposition at the places v ∉ S. Condition depends only on the double coset Ω_{v}gΩ_{v}. We can therefore assume g = a ∈ A_{v}. But then ν(K_{v} ∩gK_{v}g^{−1}) = ν(K_{v} ∩aK_{v}a^{−1}) ⊂ ν(K_{v}) contains ν(K_{v} ∩ A_{v}) = ν(K_{v}). This shows A3.

This class of elements g ∈ G(A A_{f}) and the corresponding characteristic functions of double cosets KgK generates a subalgebra H_{S,K} of the convolution algebra H of all Q-valued test functions on G(A A_{f}). It is stable under multiplication by truncation functions χ̂ factoring over the map G → G^{ab}.

\underline{Remark}: Assuming A1, A2 and A3 we even get an algebra isomorphism between the Q-Heckealgebra of some classical pair G(Q), Γ (restricted set of double cosets with respect to the discrete group Gamma) and the convolution algebra H_{S,K}. The isomorphism is

defined by ΓγΓ ↦ f(x) = vol(K)^{−1}1_{Kγ^{−1}K}(x).

If the group K is a normal subgroup of Ω, then the map f ↦ \overline{f} preserves H_{S,K}. By such choice one can construct a large class of examples where conditions A1, A2 and A3 holds for all M (attached to all Levi subgroup).

\underline{Remark}: The elliptic traces T_{ell}^{G}(f, τ) vanish for operators f corresponding to double cosets KgK, unless h_{f}γ_{f}h_{f}^{−1} ∈ KgK can be satisfied. for at least one γ ∈ G(Q). This in

turn implies ν(γ_{f}) ∈ ν(g)ν(K), hence ν(g) ∈ ν(G(Q))ν(K)ν(G(R)) (condition A2).

On the other hand, assume that g defining the Hecke operator f corresponding to KgK does not satisfy A2, then the Hecke operator f permutes the fibers of the map S_{K}(G) → G^{ab}(A A)/ν(G(Q))ν(G(R))ν(K) without fixed fibres. Hence the Lefschetz trace vanishes, because there are no fixed points. This implies, that the trace formulas of section 10 and 11 also hold without assumption A2. So for apllications of the trace formula assumption A2 can finally be dropped.

\underline{Definition}: Assume A1 holds for G. A locally constant function with compact support on G(A A_{f}) is called an \underline{allowable} Hecke operator, if it a finite linear combination of characteristic functions of double cosets KgK attached to pairs satisfying condition A3. Operators in H_{S,K} are allowable.

Appendix 2

Let G_{∞} be the group of real points of a reductive group over R. Let K be a maximal compact group, Z the connected component of the center and K̃ = Z ·K. Then for every y ∈ K̃ the set S of all x ∈ G such that x^{−1}yx ∈ K̃ is S = G_{y} · K̃. Here G_{y} denotes the centralizer of y in G. The proof is easily reduced to the case Z = 1. Namely G = ^{0}G · V, where V is the maximal vector group in the center of G_{∞}(R) and ^{0}G is a normal subgroup as in [Sp], p.19. Then K̃ = Z · K = V · K and K = K̃ ∩ ^{0}G. This reduces to the case y ∈ K and x^{−1}yx ∈ K. Choose some Cartan involution θ on G. Then x ∈ K if and only if θ(x) = x. For every solution x ∈ S the element z = θ(x)x^{−1} defines an element in G_{y} satisfying θ(z) = z^{−1}. But y, hence also G_{y} is θ-stable. Therefore there exists a symmetric one parameter subgroup in G_{y} passing through z. See e.g. [Sp] p.20. Hence we find

a symmetric root r ∈ G_{y}, θ(r) = r^{−1} of z = r^{2}. We get 1 = r^{−1}θ(x)x^{−1}r^{−1} = θ(rx)(rx)^{−1}.

Thus rx ∈ K. The group G_{y} is θ-stable. Therefore G_{y}(R) = G_{y}^{0}(R) · (K ∩ G_{y}(R)). This is true for the connected component in the topological sense or in the Zariski sense ([Sp], 5.6 (ii)). This completes the proof.

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A special Case of the Fundamental Lemma I

( The case GSp(4, F) )

R.Weissauer Mannheim, November 1993

\underline{Introduction}:

revised version In the following we are concerned with the \underline{fundamental lemma} for endoscopic groups (see [LS]) in the case, where the underlying group is the group of symplectic similitudes

G = GSp(4, F) ,

i.e. the group of all 4 × 4-matrices g satisfying

g^{t} ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) g = λ(g) ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) .

with coefficients in a local field F of positive residue characteristic different from 2. Let q be the number of elements in the residue field.

It was shown in [H], that in principal endoscopic matching of orbitral integrals holds for the group GSp(4, F). However, the proof given there does not describe the correspondence between functions with matching orbital integrals, which is relevant for many applications. Often it is important to know this correspondence for functions in the Hecke algebra. Using the trace formula one can reduce this to the study of one particular function, the unit element 1_{K} of the Hecke algebra. In the following we restrict ourselves to this important case.

There are essentially two types of tori, for which endoscopic comparison lemmas are needed in case of the group GSp(4, F). In this paper we deal with the first, the other case will be considered in a separately.

\underline{The torus T}:

In this section F can be any field of characteristic ≠ 2. Consider the subgroup H ⊂ GSp(4, F) of all matrices

(s, s′) = ( \begin{matrix}α & 0 & β & 0 \\ 0 & α′ & 0 & β′ \\ γ & 0 & δ & 0 \\ 0 & γ′ & 0 & δ′ \end{matrix} ) , s = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) , s′ = ( \begin{matrix}α′ & β′ \\ γ′ & δ′ \end{matrix} )

for s, s′ ∈ Gl(2, F), such that det(s) = det(s′). Furthermore consider the torus T

T = {(x, x′) ∈ L^{∗} × L^{∗}|Norm(x/x′) = 1} ,

for a field extension L/F of degree 2. T can be embedded as maximal torus into H and G. Any embedding is conjugate to one of the following standard form, where A = A_{0} can be assumed to be fixed such that L = F (\sqrt{A}):

Choose A′ ∈ F such that L = F (\sqrt{A}) = F (\sqrt{A′}). This means \sqrt{A}/\sqrt{A′} = α(A, A′) for some α(A, A′) ∈ F^{∗}. For simplicity we assume A, A′ to be \underline{integral}. Embed T by the A resp. A′-regular embeddings, which map given elements x = (x, x′) of T

x = a + b\sqrt{A} ∈ L , x′ = a′ + b′\sqrt{A′} .

to the pair η = (s, s′)

s = ϕ\sqrt{A}(x) = ( \begin{matrix}a & Ab \\ b & a \end{matrix} ) , s′ = ϕ\sqrt{A′}(x′) = ( \begin{matrix}a′ & A′b′ \\ b′ & a′ \end{matrix} ) .

Two such standard embeddings are conjugate (in G or H) if and only if the quotient θ = α(A, A′)/α(A, A′′) of the corresponding α’s is in Norm(L^{∗}). The second embedding is

ϕ\sqrt{A}(x) = ( \begin{matrix}a & Ab \\ b & a \end{matrix} ) , ϕ\sqrt{A′′}(x′) = ( \begin{matrix}a′ & A′θb′ \\ θ^{−1}b′ & a′ \end{matrix} ) .

\underline{Problem}:

Now assume F to be a local field with residue characteristic different from two. Our aim is to calculate the GSp(4, F) orbital integral

O_{η}^{G}(1_{K}) = ∫_{G_{η}\G} 1_{K}(g^{−1}ηg)dg/dg_{η}

for the \underline{unit element} 1_{K} of the Hecke algebra (= characteristic functions of the unimodular symplectic similitudes). This will be used to calculate the so called κ-orbital integrals in the end:

O_{η}^{κ}(1_{K}) = O_{η}^{G}(1_{K}) − O_{η′}^{G}(1_{K}) ,

where

η′ = (ϕ\sqrt{A_{0}}(x), ϕ_{θ\sqrt{A′_{0}}}(x′)) , η = (ϕ\sqrt{A_{0}}(x), ϕ\sqrt{A′_{0}}(x′)) ,

and where θ ∈ F^{∗} is such that F^{∗} = Norm(L^{∗}) ∪ θNorm(L^{∗}).

We furthermore assume \underline{regularity} x, x′ ≠ 1 and x′ ≠ x, x^{σ}, where σ denotes the nontrivial automorphism of L/F.

\underline{Assumption}: Choose A_{0}, A′_{0} to be normalized of minimal order 0 or 1. However this does not mean, that A′ = A′_{0}θ^{2} is also normalized, because in the unramified case ord(θ) = 1.

\underline{The endoscopic group M}:

As the torus T is the pullback of the maps Norm : L^{∗} × L^{∗} → F^{∗} × F^{∗} and the diagonal embedding F^{∗} → F^{∗} × F^{∗}, the dual group T̂ is the pushout of the the diagonal map C^{∗} × C^{∗} → (C^{∗})^{2} × (C^{∗})^{2} and the multiplication map C^{∗} × C^{∗} → C^{∗}. This is T̂ = (C^{∗})^{2} × (C^{∗})^{2}/{(t, t, t^{−1}, t^{−1})}. The galois group acts by the generator σ of Gal(L/F) permuting the first and second and third and fourth component respectively. The map

(z_{1}, z_{2}, z_{3}, z_{4})/(t, t, t^{−1}, t^{−1}) ↦ (z_{1}z_{4}, z_{2}/z_{1}, z_{3}/z_{4}) induces an isomorphism T̂ ≅ (C^{∗})^{3}. The

action of σ is transported to

σ(x, ν, μ) = (xνμ, ν^{−1}, μ^{−1}) .

This defines an embedding of L-groups

ψ : ^{L}T → ^{L}GSp(4, F) = GSp(4, C) × Γ

by ψ(x, ν, μ) = diag(x, xν, xμν, xμ) and

ψ(σ) = ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) × σ .

Γ can not be taken Gal(L/F), but Gal(\overline{F} /F). Identify T̂ with its image. Then Z(Ĝ) is the

group of all (x, ν, μ) = (x, 1, 1) and

T̂^{Γ} = Z(Ĝ) ∪ Z(Ĝ)κ ,

where κ = (1, −1, −1). The centralizer of ψ(κ) = diag(1, −1, 1, −1) is

M̂ = M̂^{0} = \{( \begin{matrix}∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \\ ∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \end{matrix} ) ⊂ GSp(4, C)\} .

As ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) is contained in this group, the general construction defines the trivial

homomorphism ρ : Γ → Out(M̂). We therefore consider ^{L}M = M̂ × Γ as a subgroup of LGSp(4, F) in the trivial way. The morphism ψ factors in the evident way

ψ : ^{L}T → ^{L}M ⊂ ^{L}GSp(4, F) .

The group M = Gl(2, F)^{2}/F^{∗}

M = Gl(2, F) × Gl(2, F)/{(t, t, t^{−1}, t^{−1}) | t ∈ F^{∗}}

has ^{L}M as its L-group. Namely X^{∗}(T_{M}) = {n_{1}x_{1}^{∗} + n_{2}x_{2}^{∗} + n_{3}x_{3}^{∗} + n_{4}x_{4}^{∗} | n_{1} + n_{2} = n_{3} + n_{4}}

and X_{∗}(T_{M}) = ZZx_{1∗} + .. + ZZx_{4∗}/ZZ(x_{1∗} + x_{2∗} − x_{3∗} − x_{4∗}) with the obvious roots and coroots ±(x_{1}^{∗} − x_{2}^{∗}), ±(x_{3}^{∗} − x_{4}^{∗}) resp. ±(x_{1∗} − x_{2∗}), ±(x_{3∗} − x_{4∗}). Dualizing gives the root system of

the maximal torus in the group {(g_{1}, g_{2}) ∈ Gl(2, C)| det(g_{1}) = det(g_{2})}.

Consider some torus

T_{M} = (L^{∗} × L^{∗})/F^{∗} ⊂ M = (Gl(2, F)^{2})/F^{∗} .

The isomorphism

ρ : (L^{∗} × L^{∗})/F^{∗} ≅ L^{∗} × (L^{∗}/F^{∗}) ≅ L^{∗} × N^{1}(L) ≅ (L^{∗} × L^{∗})^{0} ≅ T ⊂ GSp(4, F) ,

given by the maps

(t_{1}, t_{2})/(t^{−1}, t) ↦ (t_{1}t_{2}, t_{2} mod t) ↦ (t_{1}t_{2}, t_{2}^{σ}/t_{2}) ↦ (t_{1}t_{2}, t_{1}t_{2}(t_{2}^{σ}/t_{2})) ↦ (ϕ_{A}(t_{1}t_{2}), ϕ_{A′}(t_{1}t_{2}^{σ})) ,

defines an admissable embedding of the torus

T_{M} = (L^{∗} × L^{∗})/F^{∗} → T ⊂ GSp(4, F)
(t_{1}, t_{2}) mod (t^{−1}, t) ↦ η = (ϕ_{A}(x), ϕ_{A′}(x′)) ,

where x = t_{1}t_{2} and x′ = t_{1}(t_{2})^{σ}.

Let us record for comparison with [LS] the following formulas for D_{G}(η) = ∏_{α} |α(η) − 1|^{1/2} (product over all roots)

D_{G}(η) = |1 − x′/x||1 − x/x^{σ}||1 − x′/(x′)^{σ}||1 − x/(x′)^{σ}|

and similar

D_{H}(t_{1}, t_{2}) = |1 − t_{1}/t_{1}^{σ}||1 − t_{2}/t_{2}^{σ}| = |1 − x/(x′)^{σ}||1 − x/x′| .

\underline{Orbital integrals on M}:

Observe that the (stable) orbital integrals SO_{(t_{1},t_{2})}^{M}(1_{K}) on M are the same as the orbital integrals on Gl(2, F) × Gl(2, F), up to some normalization of measures

∫_{T \M} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dt = [o_{T} : ρ((o_{L}^{∗})^{2})] ∫_{(L^{∗}\Gl(2,F))^{2}/F^{∗}} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dl .

Here and in the following 1_{K} always denotes the unit element of the Hecke algebra of the corresponding group under consideration. We have

∫_{Gl(2,F)/L^{∗}} 1_{K}(g^{−1}t_{i}g)dg/dl = 1_{o^{∗}(L)}(t_{i})(−\frac{1}{q − 1} + \frac{q|A_{0}|^{1/2}}{q − 1} |t_{i} − t_{i}^{σ}|^{−1})

in the case of \underline{ramified} extensions L/F and

∫_{Gl(2,F)/L^{∗}} 1_{K}(g^{−1}t_{i}g)dg/dl = 1_{o^{∗}(L)}(t_{i})(−\frac{2}{q − 1} + \frac{|A′_{0}|^{1/2}(q + 1)}{q − 1}|t_{i} − t_{i}^{σ}|^{−1})

in the case of \underline{unramified} extensions L/F.

We will show by explicit computations

\underline{Theorem}: For residue characteristic different from 2 we have

Δ(η, (t_{1}, t_{2}))O_{η}^{κ}(1_{K}) = SO_{(t_{1},t_{2})}^{M}(1_{K}) ,

where

Δ(η, (t_{1}, t_{2})) = (χ_{L/F} | . |)((x − x^{σ})(x′ − (x′)^{σ}))|xx′|^{−1}

is a transfer factor. Here χ_{L/F} denotes the quadratic character of F^{∗} attached to the field extension L/F. Furthermore η = (ϕ_{A_{0}}(x), ϕ_{A_{0}}(x′)) ∈ T with ”image” (t_{1}, t_{2}) ∈ T_{M}

under ρ^{−1} and x = t_{1}t_{2}, x′ = t_{1}t_{2}^{σ}.

\underline{Reduction to H} :

In the following π and Π denote prime elements of F respectively L. Using M.Schröder’s double coset representatives [S] for H \ G/K and appendix 2, we are led to calculate an infinite (actually then finite) sum

O_{η}^{G}(1_{K}) = ∑_{i=0}^{∞} \frac{vol_{G}(K)}{vol_{H}(K_{i})}O_{η}^{H}(1_{K_{i}}) ,

where

K_{i} = {(h_{1}, h_{2}) ∈ K_{H} = Gl(2, o) | h_{1} = h_{2}^{σ} mod π^{i}, det(h_{1}) = det(h_{2})} .

Here o = o_{F} and h^{σ} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) h ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), inducing σ on ϕ_{A′}(L^{∗}). Finally

O_{η}^{H}(1_{K_{i}}) = ∫_{T \H} 1_{K_{i}}(h^{−1}ηh)dh/dt .

Here integration is over all elements in H

h = (h_{1}, h_{2}) , det(h_{1}) = det(h_{2}) .

The same formula holds for the κ-orbital integrals.

\underline{Remark}: Here we used slightly better representatives then in [S], p.106-107.

We can and will use measures on G, H and G_{η} = T, such that the volumes of maximal compact subgroups are one, i.e. vol_{G}(K) = 1, vol(K_{H}) = 1 and vol(o_{T}) = 1. This means

vol_{G}(K)/vol_{H}(K_{i}) = [K_{0} : K_{i}] .

\underline{Preliminary considerations on (x, x′)}:

Fix some (x, x′) ∈ T. The coordinates a, b and a′, b′ of x and x′ depend on the choice of the parameters A and A′. From now on we fix notation, such that these coordinates will be reserved for some fixed choice of normalized A_{0} and A′_{0}.

Suppose x and x′ are \underline{units} in L^{∗} with x = x′ mod Π (i.e. later χ > 0) in case L/F is ramified. Then

x = t_{1}t_{2} , x′ = t_{1}t_{2}^{σ}

is solvable with \underline{units} t_{1}, t_{2} in L^{∗}. Namely the image of o_{L}^{∗} under t ↦ t/t^{σ} is N_{L}^{1} ∩ (1 + Πo_{L}) in the ramified case and N_{L}^{1} in the unramified case. Put

χ = ord(a − a′) , f = ord(b) , f′ = ord(b′) , F = min(f, f′) .

Then b = π^{f}B, b′ = π^{f′}B′ for units B, B′ in o^{∗} and

q^{f+f′} = |\frac{x − x^{σ}}{\sqrt{A_{0}}} |^{−1}|\frac{x′ − (x′)^{σ}}{\sqrt{A′_{0}}} |^{−1} .

Furthermore

χ = ord(T r(x) − T r(x′)) = ord(t_{1}t_{2} + t_{1}^{σ}t_{2}^{σ} − t_{1}t_{2}^{σ} − t_{1}^{σ}t_{2}) = ord((t_{1} − t_{1}^{σ})(t_{2} − t_{2}^{σ}))

and

|t_{2} − (t_{2})^{σ}| = |x − x′| = |(a − a′)^{2} + (b − b′)^{2}A_{0}|^{1/2}

respectively

|t_{1} − (t_{1})^{σ}| = |x − (x′)^{σ}| = |(a − a′)^{2} + (b + b′)^{2}A′_{0}|^{1/2} .

\underline{Lemma 1}: Suppose χ > 0 if L/F is unramified. Then the following holds under the assumptions on x, x′ made above:

1) {|A_{0}|^{1/2}|t_{1} − t_{1}^{σ}|^{−1}, |A′_{0}|^{1/2}|t_{2} − t_{2}^{σ}|^{−1}} = {q^{F} , q^{χ−F −ord(A_{0})}}
2) χ = ord(A_{0}(b^{2} − (b′)^{2})) , hence
3) χ ≥ ord(A_{0}) + 2F with equality in case f ≠ f′.

\underline{Proof}: Observe that χ > 0 is automatic in the ramified case by our assumptions on x and x′. We have

2χ = ord[(a − a′)^{2} − (b − b′)^{2}A_{0})] + ord[(a − a′)^{2} − (b + b′)^{2}A_{0}] .

For ord(b ± b′) ≥ ord(a − a′) this implies ord(b ∓ b′) = 0 and ord(b ± b′) = ord(a − a′) = χ > 0,

hence F = ord(b) = ord(b′) = 0. Claim 1) and 2) follow immediately. Otherwise, if for both signs ord(b ± b′) < ord(a − a′), the equality above gives

2χ = ord(b + b′)^{2} + ord(b − b′)^{2} = ord(A_{0}(b^{2} − (b′)^{2})^{2} ,

hence 2). This and F = min(ord(b), ord(b′)) = min(ord(b − b′), ord(b + b′)) implies 1).

\underline{The rings R = o/π^{i}o}:

Let R = o/π^{i}o denote the residue ring and let A, A′ be elements in R. Suppose we are given matrices s, s′ in Gl(2, R)

s = ( \begin{matrix}a & bA \\ b & a \end{matrix} ) , s′ = ( \begin{matrix}a′ & b′A′ \\ b′ & a′ \end{matrix} ) .

(By abuse of language b ∈ R stands for what later will be residue classes of elements b/π^{j} ∈ o). Let the group of invertible elements a^{2} − b^{2}A for a, b in R be

N_{A} ⊂ R^{∗} .

Then N_{A} = (R^{∗})^{2} for A ∉ R^{∗} and N_{A} = R^{∗} for A ∈ R^{∗}.

\underline{Lemma 2}: The centralizer Gl(2, R)_{s} of s in Gl(2, R) is either Gl(2, R) (if b = 0 in R) or the group of all matrices

y = ( \begin{matrix}u & vA + Ann(b) \\ v & u + Ann(b) \end{matrix} ) .

Furthermore: Such y ∈ Gl(2, R)_{s} have unique decompositions into a product of two invertible matrices of the form

y = ( \begin{matrix}u & vA \\ v & u \end{matrix} ) ( \begin{matrix}1 & g \\ 0 & 1 + g′ \end{matrix} ) , g, g′ ∈ Ann(b) .

\underline{Warning}: This does not define a semidirect product decomposition of Gl(2, R)_{s}!

\underline{Proof}: One direction is clear from

( \begin{matrix}a & bA \\ b & a \end{matrix} ) ( \begin{matrix}1 & g \\ 0 & 1 + g′ \end{matrix} ) = ( \begin{matrix}a & ag + bA \\ b & a + ag′ \end{matrix} ) = ( \begin{matrix}1 & g \\ 0 & 1 + g′ \end{matrix} ) ( \begin{matrix}a & bA \\ b & a \end{matrix} ) .

The other follows during the proof (using y ∈ Gl(2, R)) of

\underline{Lemma 3}: s and s′ are conjugate by an element y in Sl(2, R)

y^{−1}sy = s′ ∈ Gl(2, R) ,

if ond only if they are conjugate in Gl(2, R) under some matrix

y = ( \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} ) , ϵ ∈ N_{A} ⊂ R^{∗} .

In other words: a = a′, b′ = b/ϵ,b′A′ = bAϵ for some ϵ ∈ N_{A}.

\underline{Proof}: One direction is clear using conjugation with the elements

y = g^{−1} ( \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} ) , ϵ ∈ R^{∗}

in Sl(2, R), for suitable chosen elements g ∈ Gl(2, R)_{s} with det(g) = ϵ.

For the other direction we can assume b′|b by symmetry. The case b′ = 0 is trivial, so we can assume b′ ≠ 0 in R. Then y^{−1}sy = s′ implies T r(s) = 2a = 2a′ = T r(s′). By assumption

2 ∈ R^{∗}, hence a = a′ . Let y be y = ( \begin{matrix}u & v \\ z & w \end{matrix} ), then sy = ys′ reads vb′ = zAb ,ub′A′ = wbA,

wb′ = ub, zb′A′ = vb. Therefore v = z(Ab/b′) + f for some f ∈ Ann(b′) and w = u(b/b′) + f′

for some f′ ∈ Ann(b′). This implies

y = ( \begin{matrix}u & z(Ab/b′) + f \\ z & u(b/b′) + f′ \end{matrix} ) .

Now b′ ≠ 0 implies Ann(b′) ⊂ πR, hence det(y) = 1 gives (u^{2} − Az^{2})(b/b′) ∈ (1 + Ann(b′)) ⊂

(R^{∗})^{2}. This proves

ϵ = b/b′ ∈ N_{A} ⊂ R^{∗}

and u^{2} − Az^{2} ∈ R^{∗}. This gives for some new f, f′ ∈ Ann(b′)

y = ( \begin{matrix}u & zA + f \\ z & u + f′ \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} ) .

As the first term is in the centralizer of s we are done. We get the (unique) product decomposition of the elements

( \begin{matrix}u & zA + f \\ z & u + f′ \end{matrix} ) = ( \begin{matrix}u & zA \\ z & u \end{matrix} ) ( \begin{matrix}1 & g \\ 0 & 1 + g′ \end{matrix} )

for some g, g′ ∈ Ann(b′), as promised in lemma 2, by solving the equation

( \begin{matrix}u & zA \\ z & u \end{matrix} ) ( \begin{matrix}g \\ g′ \end{matrix} ) = ( \begin{matrix}f \\ f′ \end{matrix} ) .

This is possible by u^{2} − Az^{2} ∈ R^{∗}. Lemma 2 and 3 are proved. Now consider a field extension L = F (\sqrt{A}) for some integer A ∈ o. Let A ∈ R also denote its residue class. Then we deduce from lemma 2 \underline{Corollary}: In the situation of lemma 2 we have

#Sl(2, R)_{s} = #Sl(2, R) , ord(b) = ν ≥ i

if b = 0 in R or in the case of \underline{ramified} L/F

#(Sl(2, R)_{s}) = 2q^{2ν+i} , ord(b) = ν < i

respectively in the case of \underline{unramified} L/F

#(Sl(2, R)_{s}) = \{ \begin{matrix}2q^{2ν+i} & ord(A) > 0 \\ (q + 1)q^{2ν+i−1} & ord(A) = 0 \end{matrix} , ord(b) = ν < i .

\underline{Proof}: For ord(b) < i the number Sl(2, R)_{s} is

#(Ann(b)) · #({x ∈ Image(ϕ_{A}(o_{L}^{∗})) | Norm(x) ∈ 1 + Ann(b)})

by lemma 2. But Ann(b) = (π^{i−ν}) mod π^{i} has order q^{ν}. Furthermore Norm(o_{L}^{∗}) = N_{A} and the index of 1 + Ann(b) in R^{∗} is (q − 1)q^{i−ν−1}. So

#Sl(2, R)_{s} = \frac{[R^{∗} : N_{A}]}{(q − 1)}q^{2ν+1−i}#Image(ϕ_{A} : o_{L}^{∗} → Gl(2, R)) .

But ϕ_{A}(o_{L}^{∗}) has order (q − 1)q^{2i−1} for π|A and order (q^{2} − 1)q^{2i−2} for ord(A) = 0. Hence the claim follows.

\underline{On T \ H-integration}: In this section assume A_{0}, A′_{0} and θ to be normalized of order 0 or 1. Consider regular elements

η = (ϕ_{A_{0}}(x), ϕ_{θ^{2}A′_{0}}(x′))

with centralizer T. We write H as a disjoint union of double cosets

⋃_{r∈R} T · r · K_{0}

where the representatives r are of the form

r ∈ (ϕ_{A_{0}}(L∗), ϕ_{θ^{2}A′_{0}}(L^{∗}))(( \begin{matrix}1 & 0 \\ 0 & π^{j} \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & π^{j′−ord(θ)}ϵ_{0} \end{matrix} ))

for certain j, j′ ∈ ZZ and ϵ_{0} ∈ o^{∗}. This implies

∫_{H} f(h)dh = ∑_{r}[o_{T} : (T ∩ rK_{0}r^{−1})] ∫_{T} dt ∫_{K_{0}} f(trk)dk ,

hence

∫_{T \H} f(h)dh/dt = ∑_{r}[o_{T} /(T ∩ rK_{0}r^{−1})] ∫_{K_{0}} f(rk)dk .

This allows to calculate the orbital integrals on H by integrations over K_{0} and gives

O_{η}^{G}(1_{K}) = ∑_{i≥0} ∑_{r∈R}[o_{T} /(T ∩ rK_{0}r^{−1})][K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}r^{−1}ηrk)dk .

Observe a′ + b′\sqrt{A′_{0}} = a′ + (b′/θ)\sqrt{θ^{2}A′_{o}}, hence for Θ = θ/π^{ord(θ)}

η_{r} := r^{−1}ηr = (ϕ_{A_{0}π^{2j}}(x), ϕ_{Θ^{2}A′_{0}π^{2j′} ϵ_{0}^{2}}(x′))

because of

( \begin{matrix}1 & 0 \\ 0 & π^{ord(θ)−j′}ϵ_{0}^{−1} \end{matrix} ) ( \begin{matrix}a′ & b′θA′_{0} \\ b′θ^{−1} & a′ \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & π^{j′−ord(θ)}ϵ_{0} \end{matrix} ) = ( \begin{matrix}a′ & \frac{b′}{Θϵ_{0}π^{j′}} (Θϵ_{0}π^{j′})^{2}A′_{0} \\ \frac{b′}{Θϵ_{0}π^{j′}} & a′ \end{matrix} ) .

Therefore T ∩ rK_{0}r^{−1} ⊂ o_{T} = {(x, x′) ∈ o_{L}^{∗} × o_{L}^{∗} | Norm(x/x′) = 1} is the group

\{(x, x′) = (a + b\sqrt{A_{0}}, a′ + b′\sqrt{A′_{0}}) | Norm(x) = Norm(x′) ∈ o^{∗}; a, a′, π^{−j}b, π^{−j′}b′ ∈ o\} .

It does not depend on the choice of the (normalized) A_{0} resp. A′_{0} resp. θ.

\underline{Notation}: Define orders o_{L}(j) = o + oπ^{j}\sqrt{A_{0}} ⊂ o_{L} for j > 0, then

\underline{Constraints (for fixed r)}: (x, x′) ∈ T ∩ rK_{0}r^{−1} means x ∈ o_{L}(j) and x′ ∈ o_{L}(j′).

Furthermore the \underline{index} is

[o_{T} /(T ∩ rK_{0}r^{−1})] = [o_{L}^{∗} : o_{L}(j)^{∗}][o_{L}^{∗} : o_{L}(j′)^{∗}][Norm(o^{∗})/Norm(o_{L}(min(j, j′))^{∗}]^{−1} .

These constraints have also to be satisfied for the nonvanishing of the orbital integral

[K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}η_{r}k)dk .

Namely k^{−1}η_{r}k ∈ K_{i} implies η_{r} ∈ K_{0}, hence (x, x′) ∈ T ∩ rK_{0}r^{−1}. Suppose the r-constraints are satisfied. Then evaluating the orbital integral

[K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}η_{r}k)dk , η_{r} = (s_{r}, (s_{r})′)

means counting all cosets kK_{i} ⊂ K_{0}, for which k^{−1}η_{r}k ∈ K_{i}. This is the number of all elements y ∈ Sl(2, o/π^{i}o), such that

y^{−1}s_{r}y ≡ (s′_{r})^{σ} mod π^{i} , y ∈ Sl(2, o/π^{i}o)

holds (a problem solved in lemma 3). It is either zero or equal to the order of the

\underline{centralizer} Sl(2, o/π^{i}o)_{s_{r}}.

\underline{Double cosets in H}: For normalizated A_{0} (resp. A′_{0}) we have

Gl(2, F) = ⋃_{j≥0} ϕ_{A_{0}}(L^{∗}) ( \begin{matrix}1 & 0 \\ 0 & π^{j} \end{matrix} ) Gl(2, o) .

This gives decompositions (with \underline{unique} integers j ≥ 0 and j′ ≥ 0) for elements (h, h′) ∈ H

h = ϕ_{A_{0}}(l) ( \begin{matrix}1 & 0 \\ 0 & π^{j} \end{matrix} ) k k ∈ Gl(2, o), l ∈ L^{∗}

h′ = ϕ_{θ^{2}A′_{0}}(l′) ( \begin{matrix}1 & 0 \\ 0 & π^{j′−ord(θ)} \end{matrix} ) k′ k′ ∈ Gl(2, o), l′ ∈ L^{∗} .

From det(h) = det(h′) we conclude

det(k)/det(k′) ∈ Norm(L^{∗})π^{j′−j−ord(θ)} .

In the \underline{ramified} case we choose once and for all π = −A_{0}, such that π = Norm(Π). Then det(k)/det(k′) ∈ Norm(L^{∗}) ∩ o^{∗} = (o^{∗})^{2}. Changing l by some unit in o^{∗} allows to assume det(k) = det(k′), i.e. (k, k′) ∈ K_{0} ⊂ H. We get H as a disjoint union of double cosets

⋃_{r∈R} T · r · K_{0} , R = N_{0} × N_{0}

and

r = r_{j,j′} = (ϕ_{A_{0}}(Π^{−j}), ϕ_{A′_{0}}(Π^{−j′}))(( \begin{matrix}1 & 0 \\ 0 & π^{j} \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & π^{j′} \end{matrix} )) .

Now consider the \underline{unramified} case. Here we get first of all π^{j′−j−ord(θ)} ∈ Norm(L^{∗})o^{∗} or

j′ − j − ord(θ) = 0 mod 2 .

Secondly, to achieve det(k) = det(k′), we are only allowed to change k or k′ by elements in

ϕ_{A_{0}π^{2j}}(L^{∗}) ∩ Gl(2, o) respectively ϕ_{θ^{2}A′_{0}π^{2j′}} (L^{∗}) ∩ Gl(2, o). So det(k)/det(k′) can be changed

within the group Norm(o_{L}^{∗}(j)o_{L}^{∗}(j′)). This is (o^{∗})^{2} unless min(j, j′) = 0. We get H as a disjoint union of double cosets

⋃_{r∈R} T · r · K_{0}

where R ⊂ ((o^{∗}/(o^{∗})^{2}) × N × N) ∪ (1 × N_{0} × 0) ∪ (1 × 0 × N) consists of all elements (ϵ_{0}, j, j′) ∈

(o^{∗}/(o∗)^{2}) × N^{2} with the property

j − j′ − ord(θ) = 0 mod 2 .

The representatives are

r = r_{ϵ_{0},j,j′} = (ϕ_{A_{0}}(Π^{−[j/2]}), ϕ_{θ^{2}A′_{0}}(Π^{−[j′−ord(θ)/2]}ϵ′_{0}))(( \begin{matrix}1 & 0 \\ 0 & π^{j} \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & π^{j′−ord(θ)}ϵ_{0} \end{matrix} )) .

Here ϵ′_{0} ∈ o_{L}^{∗} is chosen, such that Norm(ϵ′_{0}) = ϵ_{0}^{−1}, where ϵ_{0} ∈ o^{∗} \ (o^{∗})^{2}.

\underline{Summation conditions}:

Let us use the notations F = min(f, f′), ν = f − j , ν′ = f′ − j′, χ = ord(a − a′). Then

b = Bπ^{ν}, b′ = B′π^{ν′} for some units B, B′ ∈ o^{∗}. Put A_{0} = A′_{0} and θ = Θπ^{ord(θ)}. If L/K is unramified we put θ = π. From lemma 3 we get conditions on the summation indices (i, r) ∈ N_{0} × R, in order to yield a nonzero contribution to the orbital integral. These are:

1) 0 ≤ i ≤ χ

2) ν′ ≥ i iff ν ≥ i. If ν < i then ν = ν′ has to hold, furthermore also −ΘϵB′/B ∈ N_{A} = (R^{∗})^{2} unless L/K is unramified and ν = f.

3) 2f + ord(A_{0}) − ν ≥ i iff 2f′ + ord(A_{0}) − ν′ ≥ i. If 2f + ord(A_{0}) − ν < i, then f = f′

and i ≤ χ − ν has to hold.

For condition 3) observe, that the conditions of lemma 3

(b/π^{j})π^{2j}A_{0} = −(b′/π^{j′})π^{2j′}A′_{0}Θϵ_{0}ϵ^{−1} mod π^{i}

and

(b/π^{j}) = −(b′/π^{j′}Θϵ_{0})ϵ mod π^{i}

(for some ϵ ∈ N_{A}) can be combined and thus simplified. Here ϵ_{0} denotes, changing previous notation, the representative of o^{∗}/(o^{∗})^{2}. Observe (b/π^{j})π^{2j}A_{0} = Bπ^{ν+(2f−2ν)}A_{0} = BA_{0}π^{2f−ν} and similar in the other case. Therefore 2f + ord(A_{0}) − ν < i implies 2f′ +

ord(A_{0}) − ν′, yielding 2f + ord(A_{0}) − ν = 2f′ + ord(A_{0}) − ν′. But ν ≤ 2f − ν < i and 2) gives

equivalently f = f′ plus the remaining condition

ϵ = −Θϵ_{0}B′/B mod π^{i−2f−ord(A_{0})+ν} .

This together with

ϵ^{−1} = −(Θϵ_{0})^{−1}B′/B mod π^{i−ν}

means (B′/B)^{2} = 1 mod π^{i−2f−ord(A_{0})+ν} or in other words i−2f−ord(A_{0})+ν ≤ ord(B^{2}−(B′)^{2}).

Now 2f + ord(A_{0}) − ν < i forces i > 0, which is only possible for χ = ord(a − a′) > 0 by 1). From χ > 0 we get χ = ord(A_{0}(b^{2} − (b′)^{2})) by lemma 1. Hence by lemma 1 either

χ = 2F + ord(A_{0}) (f ≠ f′)

holds or

χ = 2F + ord(A_{0}(B^{2} − (B′)^{2})) (f = f′) .

We already know f = f′. Therefore it can be further reformulated to:

2f + ord(A_{0}) − ν < i implies f = f′ and i ≤ χ − ν .

\underline{Résumé}: We now express the results of the chapter on T \ H integration in terms, which are suitable for summation. We saw, that the domain of possible values i is divided into three parts: \underline{A) Conditions on i} (apart from 0 ≤ i ≤ χ ):

\underline{Range of small i}: 0 ≤ i ≤ ν and 0 ≤ i ≤ ν′ \underline{Middle range i}: ν < i ≤ 2f + ord(A_{0}) − ν and ν′ < i ≤ 2f′ + ord(A_{0}) − ν′.

In the middle range automatically ν = ν′ holds.

\underline{Large range i}: 2f + ord(A_{0}) − ν < i ≤ χ − ν and 2f′ + ord(A_{0}) − ν′ < i ≤ χ − ν′.

In the large range automatically ν = ν′ and f = f′ holds. B) Conditions on ν, ν′ and θ: In the domain of possible values ν, ν′ we have, in addition to possible restrictions mentioned above, the conditions:

0 ≤ ν ≤ f , 0 ≤ ν′ ≤ f′
ν − ν′ = f − f′ + ord(θ) mod 2 .

Furthermore in the middle and large range (concerning i) we have contributions only if

−θϵB′/B ∈ (o^{∗})^{2}

is satisfied, at least unless both ν = f and L/F is unramified!

\underline{C) Contributions from the index index(ν, ν′)}: The relevant contributions are

\frac{q + 1}{q}q^{f+f′−ν−ν′} ν = f, ν′ ≠ f′ or ν′ = f′, ν ≠ f

\frac{1}{2} \frac{(q + 1)^{2}}{q^{2}} q^{f+f′−ν−ν′} ν < f, ν′ < f′ ,

1 (ν, ν′) = (f, f′)

in the \underline{unramified} case and

q^{f+f′−ν−ν′}

in the \underline{ramified} case. (See appendix).

\underline{D) Contributions from the centralizer centralizer(ν, i)}:

(q^{2} − 1)q^{3i−2} for i ≥ 1 , else 1 (i ≤ ν small range)
2q^{2ν+i} (i > ν middle and large range) ,

unless

\frac{q + 1}{q} · q^{2ν+i} (ν = f, ν′ = f′, L/F unramified) .

In principle we now could compute the orbital integrals. We leave this to the reader. We concentrate instead on the κ-orbital integrals.

\underline{The summation (ramified case)}:

Assume, η = (s, s′) satisfies x, x′ ∈ o_{L}^{∗} (otherwise our κ-orbital integral is zero). For the κorbital integral, we do not get any contribution from the small range of the i-summation in the \underline{ramified} case, because this contribution is stable (does not depend on θ). The middle and large range contributions remain. The whole summation is empty unless χ > 0, i.e. x = x′ mod Π in addition to x, x′ ∈ o_{L}^{∗}. This is true, because χ = 0 implies i = 0, which is in the stable range. This allows to apply the preliminary remarks, made earlier.

Let χ_{L/F} be the quadratic character of F^{∗} attached to L/F. It is trivial on (o^{∗})^{2} with

χ_{L/F} (θ) = −1. We get

O_{η}^{κ}(1_{K}) = (2q^{f+f′}χ_{L/F} (−B/B′)) ∑_{0≤ν≤F} ∑_{ν<i≤χ−ν} q^{i} .

or more precisely

O_{η}^{κ}(1_{K}) = 2q^{f+f′}χ_{L/F} (−B/B′) · 1_{O_{L}^{∗}×(1+ΠO_{L})}(x, x/x′) ∑_{0≤ν≤F} ∑_{ν<i≤χ−ν} q^{i} .

The double sum gives

(q − 1)^{−1} ∑_{0≤ν≤F}(q^{χ+1−ν} − q^{ν+1})

= (q − 1)^{−1}[q^{χ+1}(q^{−F −1} − 1)/(q^{−1} − 1) − q(q^{F +1} − 1)/(q − 1)]

= \frac{q}{(q − 1)^{2}} (1 − q^{F +1})(1 − q^{χ−F +1}) = q ∫_{Gl(2,F)^{2}/(L^{∗})^{2}} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dl .

Remember b = π^{f}B, b = (x − x^{σ})/2\sqrt{A} and π ∈ Norm(L^{∗}). Hence O_{η}^{κ}(1_{K}) is equal to

2|\frac{x − x^{σ}}{2\sqrt{A_{0}}} |^{−1}|\frac{x′ − (x′)^{σ}}{2\sqrt{A_{0}}} |^{−1}χ_{L/F} (\frac{x^{σ} − x}{2\sqrt{A_{0}}})χ_{L/F} (\frac{x′ − (x′)^{σ}}{2\sqrt{A_{0}}} ) · q ∫_{Gl(2,F)^{2}/(L^{∗})^{2}} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dl .

This implies the theorem in the case L/F ramified. The factor 2 disappears, because in the ramified case [o_{T} : ρ((o_{L}^{∗})^{2})] = 2. Furthermore |A_{0}|q = 1 and χ_{L/F} (A_{0}) = χ_{L/F} (−1), because −A_{0} = Norm_{L/F} (\sqrt{A_{0}}) and |xx′| = 1 on the support.

\underline{The summation (unramified case)}:

To compute the κ-orbital integral we concentrate on the case χ > 0. The case χ = 0 is very simple, because only the small range summation contributes with the value (−q)^{f+f′}. Let us therefore turn to the essential case χ > 0. From the résumé we get the following contributions:

1) \underline{Small range}: Its contribution will be

SRC = ∑_{0≤ν≤f} ∑_{0≤ν′≤f′} ∑_{0≤i≤min(ν,ν′)} index(ν, ν′)#centralizer(i)

= ∑_{0≤i≤F} #centralizer(i)(( ∑_{i≤ν<f} ∑_{i≤ν′<f′} \frac{1}{2} · 2 · \frac{(q + 1)^{2}}{q^{2}} (−q)^{f+f′−ν−ν′}) + ( ∑_{i≤ν′<f′} \frac{q + 1}{q}(−q)^{f′−ν′})

+( ∑_{i≤ν<f} \frac{q + 1}{q}(−q)^{f−ν}) + 1) .

The representatives ϵ appearing in R give an additional factor 2 in the first sum! So

SRC = ∑_{0≤i≤F} #centralizer(i)S(i, f, f′) ,

where #centralizer(i) = 1, (q^{2} − 1)q^{3i−2} and S(i, f, f′) = (S(i, f) + 1)(S(i, f′) + 1) with

S(i, f) = \frac{q + 1}{q} ∑_{i≤ν<f}(−q)^{f−ν} = (−q)^{f−i} − 1 .

This gives

SRC = (−q)^{f+f′}(1 + \frac{q^{2} − 1}{q^{2}} ∑_{0<i≤F} q^{i}) = (−q)^{f+f′}(1 + \frac{q + 1}{q}(q^{F} − 1)) .

2) \underline{Middle range contribution}: From ν = ν′ we get

MRC = ∑_{0≤ν≤F} ∑_{ν<i≤2F −ν}(−1)^{f+f′−2ν}index(ν) · #centralizer(ν, i) .

The summation condition forces ν < F, hence

MRC = ∑_{0≤ν<F} ∑_{ν<i≤2F −ν}(\frac{1}{2} \frac{(q + 1)^{2}}{q^{2}} (−q)^{f+f′−2ν})·(2q^{2ν+i}) = (−q)^{f+f′} \frac{(q + 1)^{2}}{q^{2}} ∑_{0≤ν<F} ∑_{ν<i≤2F −ν} q^{i}

= (−q)^{f+f′} \frac{(q + 1)^{2}}{(q − 1)^{2}q} (q^{2F +1} − q^{F +1} − q^{F} + 1) .

3) \underline{Large range contribution}: We have f = f′ = F and ν = ν′. This gives

LRC = ∑_{0≤ν≤F} ∑_{2F −ν<i≤χ−ν}(−1)^{2F +2ν}index(i) · #centralizer(ν, i)

= ∑_{0≤ν<F} ∑_{2F −ν<i≤χ−ν}(\frac{1}{2} \frac{(q + 1)^{2}}{q^{2}} q^{2F −2ν}) · (2q^{2ν+i}) + ∑_{F <i≤χ−F} 1 · \frac{q + 1}{q}q^{2F +i}

= \frac{(q + 1)(−q)^{2F}}{q} (\frac{q + 1}{q}( ∑_{0≤ν<F} \frac{q^{χ−ν+1} − q^{2F −ν+1}}{q − 1} ) + (\frac{q^{χ−F +1} − q^{F +1}}{q − 1} ))

= (−q)^{f+f′} \frac{q + 1}{(q − 1)^{2}q} ((q − 1)(q^{χ−F +1} − q^{F +1}) + (q + 1)(q^{χ+1} + q^{F +1} − q^{χ−F +1} − q^{2F +1})) .

Adding together the contributions from 1),2) and 3) gives

SRC + MRC + LRC = \frac{(−q)^{f+f′}}{(q − 1)^{2}} ((q + 1)q^{F} − 2)((q + 1)q^{χ−F} − 2) .

By lemma 1 the product over the two Gl(2, F)-orbital integrals equals

\frac{1}{(q − 1)^{2}} ((q + 1)q^{F} − 2)((q + 1)q^{χ−F} − 2) .

This proves the theorem in the unramified case. \underline{References}:

[H] Hales T.C., Shalika germs on GSp(4), asterisque 171-172, p.195 - 256 (1989)

[LS] Langlands R.P. -Shelstad D., On the definition of transfer factors, Math. Ann. 278, p. 219-271 (1987)

[S] Schröder M., Thesis (university of Mannheim), 1993

[W] Warner, ...

\underline{Appendix on orders (ramified case)}: Let the situation be as above. Then the orders o_{L}(n) ⊂ o_{L} (where n > 0 by assumption) have the following properties:

1) Norm(o_{L}(n)^{∗}) = (o^{∗})^{2} This is clear from (o^{∗})^{2} ⊂ Norm(o_{L}(n)^{∗}) ⊂ Norm(o_{L}^{∗}) ⊂ (o^{∗})^{2}.

2) [o_{L}^{∗}(n) : (1 + π^{n}o_{L})] = [o^{∗} : (o^{∗} ∩ (1 + π^{n}o_{L})] = [o^{∗} : (1 + π^{n}o)] = (q − 1)q^{n−1}

3) [o_{L}^{∗} : o_{L}^{∗}(n)] = [o_{L}^{∗} : (1 + π^{n}o_{L})]/[o_{L}^{∗}(n) : (1 + π^{n}o_{L})] = (q − 1)q^{2n−1}/(q − 1)q^{n−1} = q^{n} 4) [o_{T} : (T ∩ rK_{0}r^{−1})] = [o_{L}^{∗} : o_{L}^{∗}(j)][o_{L}^{∗} : o_{L}^{∗}(j′)] = q^{j+j′}

Finally for R = o/π^{i}o and i > 0

5) #(Sl(2, R)) = (q − 1)(q + 1)q^{3i−2}

and

6) #(Image(ϕ_{A} : o_{L}^{∗} → Gl(2, R)) = q^{i}(q − 1)q^{i−1} = (q − 1)q^{2i−1} for i > 0.

\underline{Appendix on orders (unramified case)}: Let the situation be as above. Then the orders o_{L}(n) ⊂ o_{L} have the following properties:

1) Norm(o_{L}(n)^{∗}) = (o^{∗})^{2} if n > 0 and Norm(o_{L}^{∗}) = o^{∗} This is clear from (o^{∗})^{2} ⊂ Norm(o_{L}(A)^{∗}) ⊂ Norm(o_{L}^{∗}) = (o^{∗}).

2) For n > 0 we have [o_{L}^{∗}(n) : (1 + π^{n}o_{L})] = [o^{∗} : (o^{∗} ∩ (1 + π^{n}o_{L})] = [o^{∗} : (1 + π^{n}o)]= (q − 1)q^{n−1}

3) For n > 0 we have [o_{L}^{∗} : o_{L}^{∗}(n)] = [o_{L}^{∗} : (1 + π^{n}o_{L})]/[o_{L}^{∗}(n) : (1 + π^{n}o_{L})] = (q^{2} − 1)q^{2(n−1)}/(q − 1)q^{n−1} = (q + 1)q^{n−1}.

Using 1) and 3) we get

4) [o_{T} : (T ∩ rK_{0}r^{−1})] = [o_{L}^{∗} : o_{L}(j)^{∗}][o_{L}^{∗} : o_{L}^{∗}(j′)][Norm(o^{∗}) : Norm(o_{L}^{∗}(min(j, j′))].

This value is

1 , (j = j′ = 0)

(q + 1)q^{j+j′−1} , ( exactly one of the j, j′ is zero)

\frac{1}{2}(q + 1)^{2}q^{j+j′−2} , (j, j′ > 0) .

Finally for R = o/π^{i}o and i > 0

5) #(Sl(2, R)) = (q − 1)(q + 1)q^{3i−2}

and

6) #Image(ϕ_{A} : o_{L}^{∗} → Gl(2, R)) = #\{(u, v) ∈ R | Norm(u + v\sqrt{A_{0}}) ∈ R^{∗}\}. For i > 0 this

number is

(q − 1)q^{2i−1} , (π|A)
(q^{2} − 1)q^{2i−2} , (ord(A) = 0) .

\underline{Appendix 2}:

Let G be a locally compact tdc group, H a closed subgroup and K compact open in G. Let

G = ⋃ H · r · K

be a disjoint double coset decomposition with denumberable many representatives. Let dg, dh, dk be Haar measures on G, H, K such that vol_{G}(K) = 1 and vol_{K}(K) = 1. Then according to Warner p.478

∫_{HxK} f(g)dg = const′ ∫_{H/H∩(xKx^{−1})} dh/dh_{x} ∫_{K} f(hxk)dk

= const ∫_{H}(∫_{K} f(hxk)dk)dh

The constants depend on the choice of measures dh_{x} on H ∩ (xKx^{−1}) and dh on H. Evaluating with the function f(g) = 1_{xK}(g) gives

1 = const ∫_{H} 1_{H∩xKx^{−1}}(h)dh = vol_{H}(H ∩ xKx^{−1}) .

So if one \underline{normalizes} dh such that

vol_{H}(K_{H}) = 1

then

const = vol_{H}(H ∩ xKx^{−1})^{−1} = [K_{H} : (H ∩ xKx^{−1})] .

A special Case of the Fundamental Lemma II

( The case GSp(4, F) )

R.Weissauer Mannheim, March 1993 revised in November 1993 In the following we are concerned with the endoscopic \underline{fundamental lemma} (see [LS] for the general conjecture) in the case, where

G = GSp(4, F)

is the group of all 4 × 4-matrices, such that

g^{t} ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) g = λ(g) ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) , E = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )

with coefficients in a local field F of positive residue characteristic different from 2.

It was shown in [H], that the endoscopic fundamental lemma holds for the group GSp(4, F). However, this proof does not describe the correspondence between functions with matching orbital integrals, which is important for applications. Often it is enough to know this correspondence for functions in the spherical Hecke algebra. Using the trace formula one can reduce to the study of one particular function, the unit element 1_{K} of the Hecke algebra. We restrict ourselves to this important case.

There are essentially two types of tori, for which endoscopic comparison lemmas are needed for the case of the group GSp(4, F). In this paper we deal with the generic type, the degenerate one was considered in a previous paper [W].

\underline{The torus T}:

In this section F can be any field of characteristic ≠ 2. Consider a field extension E/F of degree 4 with an involution σ, which has fixed field E^{+} such that [E : E^{+}] = 2. Let q be the number of elements in the residue field of E^{+}. Then E^{+} = F (\sqrt{A}) for some element A ∈ F^{∗}. If F is a local field, we always assume A = A_{0} to be \underline{normalized}, i.e. integral of minimal possible order.

Consider the subgroup H ⊂ GSp(4, F) of all matrices

η = ( \begin{matrix}α_{1} & α_{2}A^{−1} & β_{1} & β_{2} \\ α_{2} & α_{1} & β_{2} & β_{1}A \\ γ_{1} & γ_{2}A^{−1} & δ_{1} & δ_{2} \\ γ_{2}A^{−1} & γ_{1}A^{−1} & δ_{2}A^{−1} & δ_{1} \end{matrix} ) .

This group is isomorphic to the group

ϕ : Gl(2, E^{+})^{0} = {g ∈ Gl(2, E^{+}) | det(g) ∈ F^{∗}} ≅ H .

The isomorphism ϕ^{−1} maps the matrix η above to the matrix

s = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,

where a = α_{1} + α_{2}\sqrt{A^{−1}}, b = β_{1} + β_{2}\sqrt{A^{−1}}, c = γ_{1} + γ_{2}\sqrt{A^{−1}}, d = δ_{1} + δ_{2}\sqrt{A^{−1}}.

Consider the torus T

T = {x ∈ E^{∗}|Norm_{E/E^{+}}(x) = xx^{σ} ∈ F^{∗}} .

T can be embedded into H and G. Its image is a maximal torus denoted T_{G}. Any such embedding is conjugate to one of the following standard embeddings:

Choose D ∈ E^{+} such that E = E^{+}(\sqrt{D}). For simplicity we assume D to be \underline{integral}. Embed T by the D-regular embeddings, mapping elements

x = a + b\sqrt{D} a, b ∈ E^{+}

of T to

s = ϕ\sqrt{D}(x) = ( \begin{matrix}a & Db \\ b & a \end{matrix} ) ∈ Gl(2, E^{+})^{0} .

Via the isomorphism above, we can view T as a maximal subtorus of H and G. Two such standard embeddings are conjugate as embeddings into G or H if and only if the quotient θ = \sqrt{D′}/\sqrt{D} is in F^{∗} · Norm_{E/E^{+}}(E^{∗}).

\underline{Problem}:

Now assume F to be a local field. Our aim is to calculate the GSp(4, F) orbital integral

O_{η}^{G}(1_{K}) = ∫_{G_{η}\G} 1_{K}(g^{−1}ηg)dg/dg_{η}

for the \underline{unit element} 1_{K} of the Hecke algebra (= characteristic functions of the group K of the unimodular symplectic similitudes). Measures are normalized by vol_{G}(K) = 1 and vol_{T} (o_{T}) = 1, where o_{T} is the subgroup of all x ∈ T with x ∈ o_{E}^{∗}. By transport of structure this defines a unique Haar measure on T ≅ T_{G} ≅ G_{η}. Here we assumed \underline{regularity} in the sense that E = F (x).

This will be used to calculate the so called κ-orbital integrals: Observe, that the subgroup F^{∗}Norm_{E/E^{+}}(E^{∗}) has at most index two in (E^{+})^{∗}. We are interested only in the unstable case, where the two groups are not equal. Let us assume, that this is the case. Then

O_{η}^{κ}(1_{K}) = O_{η}^{G}(1_{K}) − O_{η′}^{G}(1_{K}) ,

where

η = ϕ ∘ ϕ\sqrt{D_{0}}(x) , η′ = ϕ ∘ ϕ_{θ\sqrt{D_{0}}}(x) , (x ∈ T) ,

and where θ ∈ (E^{+})^{∗} is such that (E^{+})^{∗} = F^{∗}Norm_{E/E^{+}}(E^{∗}) ∪ θF^{∗}Norm_{E/E^{+}}(E^{∗}).

\underline{The unstable case}: For local fields E^{∗} considered above (E^{+})^{∗} = F^{∗}·Norm_{E/E^{+}}(E^{∗}) unless E/F is a noncyclic galois extension. As we are only interested in κ-orbital integrals, we will restrict from now on to that particular case.

By assumption the residue characteristic is different from two. Therefore the noncyclic galois extensions E/F is the composite field of the three quadratic extension fields of F in \overline{F}. Exactly on of these three extension fields L, L′, E^{∗} is unramified. This implies, that exactly one of the two extensions E/E^{+}, E^{+}/F is ramified respectively unramified.

The Galois group Gal(E/F) has 4 elements 1, σ, τ and στ = τσ. Let denote

L = E^{⟨τ⟩} = F (\sqrt{D_{0}})
L′ = E^{⟨τσ⟩} = F (\sqrt{A_{0}D_{0}})

and

E^{+} = E^{⟨σ⟩}

the three fixed fields of E. If one of the extensions L/F or L′/F is unramified, we can assume that L/F is the unramified extension. Note the asymmetry of this normalization! Observe, that E/E^{+} is ramified iff E^{+}/F is unramified iff L′/F and L/F are ramified. Furthermore if E/E^{+} is unramified, then L/F is unramified (by our choice) and E^{+}/F and L′/F are ramified. Especially, L′/F is always ramified.

\underline{Assumption}: Choose D = D_{0} ∈ F^{∗} to be normalized of minimal order 0 or 1. This does not mean, that D = D_{0}θ^{2} is also normalized or in F^{∗}, because ord(θ) = 1 in case E/E^{+} is unramified. We fix prime elements π_{E^{+}} = −D_{0} resp. π_{E^{+}} = \sqrt{A_{0}} according to whether E/E^{+} is ramified or not. The valuation on F is chosen, such that |π_{F} | = q_{0}^{−1}, where q_{0} is the cardinality of the residue field of F. Let q denote the cardinality of the residue class field of E^{+}. We extend the absolute value of F to \overline{F}.

\underline{The endoscopic group M}:

The torus T is the pullback of the maps Norm : E^{∗} → (E^{+})^{∗} and the diagonal embedding K^{∗} → (E^{+})^{∗}. Therefore the dual group T̂ is the pushout of the the diagonal map (C^{∗})^{2} → (C^{∗})^{4} and the multiplication map (C^{∗})^{2} → C^{∗}. This gives T̂ = (C^{∗})^{4}/{(t, t, t^{−1}, t^{−1})}. The Galois group G(E/F) acts by the generator σ of Gal(E/E^{+}) permuting, say the first and second and third and fourth component respectively. Let τ ≠ σ be a second involution in G(E/F) with fixed field L = F (\sqrt{D_{0}}), where we assumed D_{0} ∈ F^{∗} to be normalized. Then τ acts by permuting, say the first and third and second and fourth component.

The map (z_{1}, z_{2}, z_{3}, z_{4})/(t, t, t^{−1}, t^{−1}) ↦ (z_{1}z_{4}, z_{2}z_{4}, z_{3}/z_{4}) induces an isomorphism T̂ ≅ (C^{∗})^{3}.

The action of σ is transported to

σ(t_{1}, t_{2}, t_{3}) = (t_{2}t_{3}, t_{1}t_{3}, t_{3}^{−1}) .

The action of τ is transported to

τ(t_{1}, t_{2}, t_{3}) = (t_{2}t_{3}, t_{2}, t_{1}/t_{2}) .

This defines an embedding of L-groups

ψ : ^{L}T → ^{L}GSp(4, F) = GSp(4, C) × W_{F}

by ψ(t_{1}, t_{2}, t_{3}) = diag(t_{1}, t_{2}, t_{2}t_{3}, t_{1}t_{3}) and

ψ(σ) = i ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) × σ

ψ(τ) = ( \begin{matrix}0 & 0 & i & 0 \\ 0 & 1 & 0 & 0 \\ −i & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

Identify T̂ with its image under ψ. Then Z(Ĝ) is the group of all (t_{1}, t_{2}, t_{3}) = (x, x, 1) and

T̂^{Γ} = Z(Ĝ) ∪ Z(Ĝ)κ, where κ = (1, −1, −1). The centralizer of s = ψ(κ) = diag(1, −1, 1, −1)

is

M̂ = M̂^{0} = \{( \begin{matrix}∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \\ ∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \end{matrix} ) ⊂ GSp(4, C)\} .

Consider ^{L}M = M̂ × Γ as a subgroup of ^{L}GSp(4, F) in the trivial way. The embedding is denoted ξ. This defines an endoscopic datum (M,^{L}M, s, ξ). The group M is Gl(2, F) ×

Gl(2, F)/{(t, t, t^{−1}, t^{−1}) | t ∈ F^{∗}}.

As ψ(σ), ψ(τ) are contained in ^{L}M, the morphism ψ factors in the evident way

ψ : ^{L}T → ^{L}M ⊂ ^{L}GSp(4, F) .

Consider tori T_{M} = (L^{∗} × (L′)^{∗})/F^{∗} in M = (Gl(2, F)^{2})/F^{∗}. The isomorphism

(L^{∗} × (L′)^{∗})/F^{∗} ≅ T ,

given by the map (t_{1}, t_{2})/(t^{−1}, t) ↦ t_{1}t_{2} ↦ ϕ ∘ ϕ\sqrt{D}(t_{1}t_{2}), defines an admissable embedding of the torus

ρ : T_{M} ≅ (L^{∗} × (L′)^{∗})/F^{∗} → T → T_{G} ⊂ GSp(4, F) .

Let us record for comparison with [LS] for x = t_{1}t_{2} ∈ T

D_{G}(x) = |1 − x/x^{τ} ||1 − x/x^{σ}||1 − x^{τ} /x^{τσ}||1 − x^{τ} /x^{σ}|
D_{H}(t_{1}, t_{2}) = |1 − t_{1}/t_{1}^{σ}||1 − t_{2}/t_{2}^{σ}| = |1 − x/x^{στ} ||1 − x/x^{τ} | .

\underline{Orbital integrals on M}:

Orbital integrals on M are stable

SO_{(t_{1},t_{2})}^{M}(1_{K}) = ∫_{T_{M} \M} 1_{K_{M}} (g^{−1}(t_{1}, t_{2})g)dg/dt

Here 1_{K_{M}} is the unit elements of Hecke algebra of the group K_{M} defined in [W1]. The measure on M is chosen as in [W1], on T_{M} by transport of structure from the unique Haar measure on T, such that vol_{T} (o_{T}) = 1.

We will show by explicit computations

\underline{Theorem}: Suppose t_{1} ∈ L^{∗}, t_{2} ∈ (L′)^{∗}. Assume that x = t_{1}t_{2} ∈ T is regular in the sense that E = F (x) holds. Then η = ϕ ∘ ϕ\sqrt{D_{0}}(x) is regular with image (t_{1}, t_{2}) ∈ T_{M} under ρ^{−1}. Assume that the residue characteristic of F is different from two. Then

Δ(η, (t_{1}, t_{2}))O_{η}^{κ}(1_{K}) = SO_{(t_{1},t_{2})}^{M}(1_{K}) ,

where

Δ(η, (t_{1}, t_{2})) = χ_{L/F} (A_{0})|N_{E/F} (x)|^{−1/2}(χ_{L/F} | . |)((x − x^{σ})(x^{τ} − x^{στ}))

is a transfer factor. The character χ_{L/F} of F^{∗} is the quadratic character attached to the field extension L/F.

\underline{Remark}: Instead of working with stable orbital integrals on M, it is better to work with orbital integrals on M_{1} = Gl(2, F)^{2} via the z-extension p : M_{1} → M. We get as in [W1]

SO_{(t_{1},t_{2})}^{M}(1_{K_{M}}) = [o_{T} : ρ((o_{L}^{∗} × o_{L′}^{∗})] ∫_{(L^{∗}×(L′)^{∗})\Gl(2,F)^{2}} 1_{K_{M_{1}}} (g^{−1}(t_{1}, t_{2})g)dg/dl ,

for a suitable adjusted choice of (t_{1}, t_{2}) in the cosets of M_{1}/Kern(p). Measures are chosen

such that vol_{M_{1}}(K_{M_{1}}) = 1 where K_{M_{1}} = Gl(2, 0_{F})^{2} and such that vol_{L^{∗}×(L′)^{∗}}(o_{L}^{∗} × o_{L′}^{∗}) = 1.

We will use this, tacitly assuming that (t_{1}, t_{2}) is chosen adjusted, to rewrite the stable orbital integral on M as a product of two orbital integrals on Gl(2, F): Namely of

∫_{L^{∗}\Gl(2,F)} 1_{K}(g^{−1}t_{1}g)dg/dl = 1_{o^{∗}(L)}(t_{1})(−\frac{f(L : F)}{q_{0} − 1} + \frac{|D_{0}|^{1/2}(q_{0} − 1 + f(L : F))}{q_{0} − 1}|t_{1} − t_{1}^{σ}|^{−1})

for L/F and similar for ∫_{(L′)^{∗}\Gl(2,F)} 1_{K}(g^{−1}t_{2}g)dg/dl

1_{o^{∗}(L′)}(t_{2})(−\frac{f(L′ : F)}{q_{0} − 1} + \frac{|D_{0}A_{0}|^{1/2}(q_{0} − 1 + f(L′ : F))}{q_{0} − 1}|t_{2} − t_{2}^{σ}|^{−1}) .

Here f(L/F) resp. f(L′/F) denote the degrees of the residue field extension. Here K = Gl(2, o_{F}) and measures are chosen analogous as for M_{1}.

\underline{Reduction to H} :

Using the double coset representatives for H \ G/K and [W1] appendix 2, we are lead to calculate an infinite (actually then finite) sum and get the following \underline{formula}

O_{ϕ̌(s)}^{G}(1_{K}) = ∑_{i=0}^{∞} \frac{vol_{G}(K)}{vol_{H}(K_{i})} ∫_{ϕ̌^{−1}(T)\Gl(2,E^{+})^{0}} 1_{K_{i}}(h^{−1}sh)dh/dt ,

where

K_{i} = Gl(2, o_{E^{+}}(i))^{0} ⊂ Gl(2, o_{E^{+}})^{0} = K_{0} = K_{H} .

Here o_{E^{+}}(i) denotes the \underline{order} o_{F} + π_{F}^{i} o_{E^{+}} in o_{E^{+}} for i > 0. The index ^{0} indicates, that determinants should be in o_{F}^{∗}. Furthermore ϕ̌ : Gl(2, E^{+})^{0} ≅ H is the composite of the isomorphism ϕ and some (not necessarily interior) automorphism of Gl(2, E^{+})^{0}, which allows to identify both groups (see appendix III).

The analogous formula holds for the κ-orbital integrals. Warning: η = ϕ(s) and ϕ̌(s) are stably conjugate, but not necessarily conjugate. Hence the right side of the formula computes O_{η}^{κ}(1_{K}) up to a sign. From the following description it follows, that this sign depends on the residue class of \sqrt{A}, A = A_{0} in (E^{+})^{∗}/F^{∗}Norm_{E/E^{+}}(E^{∗}). The latter group equals (E^{+})^{∗}/Norm_{E/E^{+}}(E^{∗}) using local classfield theory and our assumption E = LL′. Therefore this sign is χ_{E/E^{+}}(\sqrt{A_{0}}^{−1}) = χ_{L/F} (\sqrt{A_{0}}^{−1}\sqrt{A_{0}}^{−τ}) = χ_{L/F} (−A_{0}). See also appendix III.

\underline{Explanation and proof of formula}: We use the double coset decomposition

G = ⋃_{i=0}^{∞} H · z(i) · GSp(4, o_{F}) ,

where

z(i) = ( \begin{matrix}1 & 0 & −μ^{−1} & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

with the integers μ = μ(i) = Aπ_{F}^{i}. See [S], A.19.3(2).

z(i)^{−1}ηz(i) = ( \begin{matrix}α_{1} + γ_{1} & \frac{α_{2}+γ_{2}}{A} & \frac{β_{1}−α_{1}+δ_{1}−γ_{1}}{μ} & \frac{β_{2}+δ_{2}}{μ} \\ α_{2} & α_{1} & \frac{β_{2}−α_{2}}{μ} & \frac{β_{1}A}{μ} \\ μγ_{1} & \frac{μγ_{2}}{A} & δ_{1} − γ_{1} & δ_{2} \\ \frac{μγ_{2}}{A} & \frac{μγ_{1}}{A} & \frac{δ_{2}−γ_{2}}{A} & δ_{1} \end{matrix} )

for

ϕ^{−1}(η) = ( \begin{matrix}a & \frac{b}{μ} \\ μc & d \end{matrix} ) .

If this matrix is in GSp(4, o_{F}), then α_{1}, α_{2}, δ_{1}, δ_{2} ∈ o_{F}. Furthermore then γ_{1} ∈ o_{F} and α_{2} = −γ_{2} = −δ_{2} mod (A). Especially γ_{2} ∈ o_{F}. Furthermore

β_{1} ∈ (μ/A)
β_{2} = α_{2} = −δ_{2} mod (μ)
β_{1} − α_{1} + δ_{1} − γ_{1} = 0 mod (μ) .

Especially β_{1}, β_{2} ∈ o_{F}. Put

g = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) = ( \begin{matrix}a − b & b \\ a + c − b − d & b + d \end{matrix} ) .

Then these conditions are equivalent to g ∈ Gl(2, o_{E^{+}})^{0} together with α_{2}, δ_{2} ∈ (μ), γ_{1} ∈ (μ) and β_{1} ∈ (γ/A) and γ_{2} ∈ (A). Here we changed notation such that α = α_{1} + α_{2}\sqrt{A^{−1}}, .., δ = δ_{1} + δ_{2}\sqrt{A^{−1}}. If we replace g by

ǧ = ( \begin{matrix}1 & 0 \\ 0 & \sqrt{A} \end{matrix} )^{−1} g ( \begin{matrix}1 & 0 \\ 0 & \sqrt{A} \end{matrix} ) ,

then these conditions are equivalent to

ǧ ∈ Gl(2, o_{E^{+}}(\frac{μ}{A}))

and det(g) ∈ o_{F}^{∗}. In other words for μ = Aπ_{F}^{i}, the isomorphism

ϕ̌: Gl(2, E^{+})^{0} → H ⊂ GSp(4, F)

g ↦ ϕ(Ad_{h(μ)}(g)) , h(μ) = ( \begin{matrix}1 & 0 \\ 0 & μ \end{matrix} ) ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & \sqrt{A} \end{matrix} ) ∈ H

identifies (ϕ̌)^{−1}(H ∩ z(i)GSp(4, o_{F})z(i)^{−1}) with the subgroup K_{i} ⊂ Gl(2, o_{E^{+}})^{0}

K_{i} = Gl(2, o_{E^{+}}(i))^{0} .

We use measures on G, H and G_{η} ≅ T, such that the volumes of the following compact subgroups vol_{G}(K) = 1, vol(K_{H}) = 1 and vol_{T} (o_{T}) = 1 are one. This implies

vol_{G}(K)/vol_{H}(K_{i}) = [K_{0} : K_{i}] .

\underline{Preliminary considerations on x}:

Fix x ∈ T. Remember τ(\sqrt{D_{0}}) = \sqrt{D_{0}}, i.e. L = F (\sqrt{D_{0}}). The coordinates a, b ∈ E^{+} and

a′ = a^{τ} , b′ = b^{τ} of

x = a + b\sqrt{D_{0}}

and

x^{τ} = a′ + b′\sqrt{D_{0}}

depend on the choice of the parameter D = D_{0}. We fixed notation, such that these coordinates are reserved for some fixed choice of \underline{normalized} D_{0} ∈ F^{∗}.

\underline{Assumption:} Suppose x ∈ T is a \underline{unit} in E^{∗}. Further assume x = x^{τ} mod π_{E} (in later notation χ > 0) in case E/E^{+} is ramified.

Then

x = t_{1}t_{2}

is solvable with \underline{units} t_{1} ∈ o_{L}^{∗} and t_{2} ∈ o_{L′}^{∗}.

\underline{Proof}: The element x can be written in the form x = T_{1}T_{2} with T_{1} ∈ L^{∗} and T_{2} ∈ (L′)^{∗}.

Observe x/x^{στ} = T_{1}/T_{1}^{σ} and x/x^{τ} = T_{2}/T_{2}^{σ}.

x ∈ T ∩ o_{E}^{∗} implies x/x^{στ} ∈ N_{L}^{1} ∩ o_{L}^{∗}. Namely x ∈ T implies xx^{σ} = x^{τ} x^{στ}, which gives

x/x^{στ} ∈ E^{⟨τ⟩} = L. Furthermore N_{L/F} (x/x^{στ}) = xx^{σ}/x^{στ} x^{τ} = 1. If t_{1}/t_{1}^{σ} = x/x^{στ} has

a solution t_{1} ∈ o_{L}^{∗}, then one can change T_{1} into t_{1}. Note that T_{1}/t_{1} ∈ F^{∗}. We can therefore also change T_{2} into t_{2} = T_{2}T_{1}/t_{1} within (L′)^{∗}. Then x = t_{1}t_{2}. Because of t_{2} = x/t_{1} ∈ o_{E}^{∗} ∩ (L′)^{∗} = o_{L′}^{∗} our claim can be satisfied.

Similar one shows x/x^{τ} ∈ L′ = E^{⟨στ⟩} and N_{L′/F} (x/x^{τ}) = 1. Therefore x/x^{τ} ∈ N_{L′}^{1}. If t_{2}/t_{2}^{σ} = x/x^{τ} were solvable with some t_{2} ∈ o_{L′}^{∗}, again one shows that the claim can be satisfied.

We assumed, that the residue characteristic is odd. Therefore the image of o_{L}^{∗} under t ↦ t/t^{σ} is N_{L}^{1} ∩ (1 + π_{L}o_{L}), if L/F ramifies, and equal to N_{L}^{1} otherwise. Similar for L′. Therefore, if one of the two extensions L/F or L′/F is unramified, then the claim is varified. Otherwise both L/F and L′/F must be ramified or equivalently E/E^{+} must be ramified. But in this case our assumption shows x/x^{τ} ∈ 1 + π_{L′}o_{L′}. Therefore we can solve t_{2}/t_{2}^{σ} = x/x^{τ} with t_{2} ∈ o_{L′}^{∗}. Hence in all cases x = t_{1}t_{2} ∈ o_{L}^{∗}o_{L′}^{∗}.

\underline{Remark}: The same argument shows [o_{T} : ρ(o_{L}^{∗} × o_{L′}^{∗})] = e_{E/E^{+}}. Namely for e_{E/E^{+}} = 2 the number x = π_{L′}/π_{L} is not in o_{L}^{∗}o_{L′}^{∗}.

Because of E = F (x) we have a ≠ a′. Let the valuation on E^{+} be normalized with minimal value 1. Put

χ = ord(a − a′) , N = ord(b) = ord(b′) .

Then χ < ∞. Furthermore b = π_{E^{+}}^{N}B, b′ = π_{E^{+}}^{N}B′ for units B, B′ in o_{E^{+}}^{∗}. For the absolute values, we get

q^{N} = |\frac{x − x^{σ}}{\sqrt{D_{0}}} |^{−2} .

Furthermore

χ = ord(T r_{E/E^{+}}(x) − T r_{E/E^{+}}(x^{τ})) = ord(t_{1}t_{2} + t_{1}^{σ}t_{2}^{σ} − t_{1}t_{2}^{σ} − t_{1}^{σ}t_{2}) = ord((t_{1} − t_{1}^{σ})(t_{2} − t_{2}^{σ}))

and

|t_{2} − t_{2}^{σ}| = |t_{2} − t_{2}^{σ}||t_{1}| = |x − x^{τ} | = |(a − a′)^{2} + (b − b′)^{2}D_{0}|^{1/2}

respectively

|t_{1} − t_{1}^{σ}| = |t_{1} − t_{1}^{σ}||t_{2}| = |x − x^{τσ}| = |(a − a′)^{2} + (b + b′)^{2}D_{0}|^{1/2} .

\underline{Lemma 1}: Suppose that x ∈ o_{E}^{∗} and χ > 0 holds. Then

1) {|D_{0}|^{1/2}|t_{1} − t_{1}^{σ}|^{−1}, |D_{0}|^{1/2}|t_{2} − t_{2}^{σ}|^{−1}} = {q^{N/2}, q^{(χ−N−ord(D_{0}))/2}}
2) χ = ord(D_{0}(b^{2} − (b′)^{2})) , hence
3) χ ≥ ord(D_{0}) + 2N.

\underline{Proof}: Observe that assumptions of lemma 1 imply the assumption made above. We have

2χ = ord[(a − a′)^{2} − (b − b′)^{2}D_{0})] + ord[(a − a′)^{2} − (b + b′)^{2}D_{0}] .

For ord(b ± b′) ≥ ord(a − a′) this implies ord(b ∓ b′) = 0 and ord(b ± b′) = ord(a − a′) = χ > 0,

hence N = ord(b) = ord(b′) = 0. Claim 1) and 2) follow immediately. Otherwise, if for both signs ord(b ± b′) < ord(a − a′), the equality above gives

2χ = ord(D_{0}(b + b′)^{2}) + ord(D_{0}(b − b′)^{2}) = ord(D_{0}(b^{2} − (b′)^{2})^{2} ,

hence 2). This and N = min(ord(b − b′), ord(b + b′)) implies 1).

Finally one has the following

\underline{Parity rule}: If E/E^{+} is unramified, then ord(D_{0}) = 0 and χ is odd (because a − a′ ∈ E^{+} \ F and E^{+}/F ramifies). On the other hand ord((t_{1} − t_{1}^{σ})/2\sqrt{D_{0}}) is even (L/F is unramified) and ord((t_{2} − t_{2}^{σ})/2\sqrt{D_{0}}) is odd (by lemma 1).

\underline{The residue rings S = o_{E^{+}}/π_{F}^{i}}:

Assume i ≥ 1. Let S denote some residue ring o_{E^{+}}/π_{F}^{i} and R the corresponding ring R = o_{F} /π_{F}^{i}. Observe, that the image of the order o_{E^{+}}(i) in S is exactly the subring R of S. Obviously for A = A_{0}

S = R[X]/(X^{2} − A) = R + R\sqrt{A} ,

where \sqrt{A} = X mod (X^{2} − A). Let Gl(2, S)^{0} be the group of matrices with determinant in R^{∗}. The image of K_{i} in Gl(2, S)^{0} is Gl(2, R).

The integrals

[K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}sk)dk ,

for σ ∈ T are zero unless s ∈ T ∩ K_{0}. Let us assume that it is nonzero. Then the value of this integral is the number of left cosets kK_{i} ⊂ K_{0} for which k^{−1}sk ∈ K_{i} holds, or equivalently the number of left cosets y · Gl(2, R) ⊂ Gl(2, S)^{0}, such that

y^{−1}sy ∈ Gl(2, R) .

\underline{Assumption}: In the following suppose s is of the form

s = ( \begin{matrix}\overline{a} & \overline{b}D \\ \overline{b} & \overline{a} \end{matrix} ) ,

where \overline{a}, \overline{b} ∈ S and D_{0}|D.

\underline{Remark}: In later application \overline{a} will be the residue class of a and \overline{b} will be the residue class of (Θϵ_{0})^{−1}Bπ_{E^{+}}^{ν}.

\underline{Lemma 2}: Let e = e(E/E^{+}) be the ramification index. We will show, that the following conditions are equivalent:

1) It exists an y ∈ Gl(2, S)^{0} such that y^{−1}sy ∈ Gl(2, R).
2) It exists a unit ϵ ∈ (S^{∗})^{e} such that

( \begin{matrix}1 & 0 \\ 0 & \frac{ϵ}{ϵ^{τ}} \end{matrix} ) s ( \begin{matrix}1 & 0 \\ 0 & \frac{ϵ}{ϵ^{τ}} \end{matrix} )^{−1} = s^{τ} .

3) It exists a unit ϵ ∈ (S^{∗})^{e} such that

( \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} ) s ( \begin{matrix}1 & 0 \\ 0 & ϵ \end{matrix} )^{−1} ∈ Gl(2, R)

in other words: \overline{a} ∈ R, ϵ\overline{b} ∈ R, ϵ^{−1}\overline{b}D ∈ R.

\underline{Proof}: The statement are trivial for \overline{b} = 0. Hence let us assume \overline{b} ≠ 0. Consider a solution y ∈ Gl(2, S)^{0} of 1) or also a ”weak” solution y ∈ Gl(2, S). Remark: It is clear, that any weak solution y with

det(y) = r · det(z) ∈ R^{∗} · det(Gl(2, S)_{s}) = R^{∗} · N_{D}

can be made into a solution by replacing y by

z^{−1}y ( \begin{matrix}1 & 0 \\ 0 & r \end{matrix} )^{−1} .

If e = 1, E/E^{+} is unramified and E^{+}/F is ramified, then R^{∗}(S^{∗})^{2} = S^{∗} ⊂ R^{∗}det(Gl(2, S)_{s}) . If e = 2, E/E^{+} is ramified and E^{+}/F is unramified, then R^{∗}det(Gl(2, S)_{s}) = (S^{∗})^{2}. Hence 3) obviously implies 1). Furthermore 2) and 3) are trivially equivalent. So let us discuss the remaining implication 1) implies 2):

Of course y^{−1}sy ∈ Gl(2, R) is equivalent to y^{−1}sy = y^{−τ} s^{τ} y^{τ}, which is equivalent to

xsx^{−1} = s^{τ}

for

x = y^{τ} y^{−1} , x^{τ} = x^{−1} .

But x \underline{uniquely} determines the left coset yGl(2, R) of the (weak) solution y.

If y is a solution, then det(x) = 1 and

x = ( \begin{matrix}x_{1} & x_{2} \\ x_{3} & x_{4} \end{matrix} )

( \begin{matrix}x_{1}^{τ} & x_{2}^{τ} \\ x_{3}^{τ} & x_{4}^{τ} \end{matrix} ) = ( \begin{matrix}x_{4} & −x_{2} \\ −x_{3} & x_{1} \end{matrix} ) ,

hence

x = ( \begin{matrix}x_{1} & r_{2}\sqrt{A} \\ r_{3}\sqrt{A} & x_{1}^{τ} \end{matrix} ) r_{2}, r_{3} ∈ R .

Observe the trace condition \overline{a} = \overline{a}^{τ}. This gives the conditions x_{1}^{τ} \overline{b} = \overline{b}^{τ} x_{1}, x_{1}\overline{b}D = x_{1}^{τ} \overline{b}^{τ} D^{τ}, \sqrt{A}r_{2}\overline{b} = \sqrt{A}r_{3}\overline{b}^{τ} D^{τ}. For \overline{b}/\overline{b}^{τ} (because of \overline{b} ≠ 0 and ord(\overline{b}) = ord(\overline{b}^{τ})) one gets

\frac{\overline{b}}{\overline{b}^{τ}} x_{1}^{τ} = x_{1} + Ann_{S}(\overline{b})

\frac{\overline{b}^{τ}}{\overline{b}} x_{1}^{τ}D^{τ} = x_{1}D + Ann_{S}(\overline{b})

\frac{\overline{b}}{\overline{b}^{τ}} r_{2}\sqrt{A} = r_{3}\sqrt{A}D^{τ} + Ann_{S}(\overline{b}) .

But det(x) = 1. Hence

x_{1}x_{1}^{τ} − r_{2}r_{3}\sqrt{A}^{2} = 1 ,

hence

\frac{\overline{b}}{\overline{b}^{τ}} = (\frac{\overline{b}}{\overline{b}^{τ}} x_{1}^{τ})x_{1} − (r_{2}\sqrt{A}\frac{\overline{b}}{\overline{b}^{τ}} )r_{3}\sqrt{A}

= x_{1}^{2} + Ann_{S}(\overline{b}) − r_{3}^{2}AD^{τ} + Ann_{S}(\overline{b}) .

But Ann_{S}(\overline{b}) ⊂ π_{E^{+}}S because \overline{b} ≠ 0 and π_{F} |A_{0}D_{0}|AD^{τ}, because E/F is always ramified! Hence

\overline{b}/\overline{b}^{τ} = x_{1}^{2} mod π_{E^{+}}S .

Therefore x_{1} ∈ S^{∗} and

\overline{b}/\overline{b}^{τ} = x_{1}/x_{1}^{τ} + Ann_{S}(\overline{b}) .

One therefore finds

x_{1}x_{1}^{τ} = 1 mod π_{E^{+}}

and

x ∈ ( \begin{matrix}x_{1} & r_{3}\sqrt{A}D^{τ} \frac{x_{1}^{τ}}{x_{1}} + Ann_{S}(\overline{b}) \\ r_{3}\sqrt{A} & x_{1}^{τ} \end{matrix} ) .

However r_{2}\sqrt{A} = r_{3}\sqrt{A}\frac{\overline{b}^{τ}}{\overline{b}} D^{τ} = \frac{x_{1}}{x_{1}^{τ}} D mod Ann_{S}(\overline{b}). Therefore

x ∈ ( \begin{matrix}1 & 0 \\ 0 & \frac{x_{1}^{τ}}{x_{1}} \end{matrix} ) ( \begin{matrix}x_{1} & \frac{x_{1}}{x_{1}^{τ}} r_{3}\sqrt{A}D + Ann_{S}(\overline{b}) \\ \frac{x_{1}}{x_{1}^{τ}} r_{3}\sqrt{A} & x_{1} \end{matrix} )

⊂ ( \begin{matrix}1 & 0 \\ 0 & \frac{x_{1}^{τ}}{x_{1}} \end{matrix} ) · Gl(2, S)_{s} .

We get (2) (weak form)

xsx^{−1} = ( \begin{matrix}1 & 0 \\ 0 & \frac{x_{1}^{τ}}{x_{1}} \end{matrix} ) s ( \begin{matrix}1 & 0 \\ 0 & \frac{x_{1}^{τ}}{x_{1}} \end{matrix} )^{−1} = s^{τ} .

Here x_{1} ∈ S^{∗}. Hence 3) (weak form)

xsx^{−1} = ( \begin{matrix}1 & 0 \\ 0 & x_{1} \end{matrix} )^{−1} s ( \begin{matrix}1 & 0 \\ 0 & x_{1} \end{matrix} ) ∈ Gl(2, R) .

It only remains to show

x_{1} ∈ (S^{∗})^{e}

in the case e(E/E^{+}) = 2. But if E/E^{+} is ramified (E^{+}/F is unramified) we have N_{D} = (S^{∗})^{2} and R^{∗} ⊂ (S^{∗})^{2}. But we have shown x_{1}x_{1}^{τ} = 1 mod π_{E^{+}}. The norm map of the residue field extension of E^{+}/F is surjective. Therefore the kernel is contained in a subgroup, whose quotient has a index-2 subgroup, thus it is contained in the group of squares. This implies

x_{1} ∈ (S^{∗})^{2}

if e = 2. Lemma 2 is proved.

In the situation of lemma 2 consider the set of solutions S

{y ∈ Gl(2, S)^{0}/Gl(2, R) | y^{−1}sy ∈ Gl(2, R)}

and the set of solutions C

{x ∈ Sl(2, S) | x^{τ} = x^{−1}, xsx^{−1} = s^{τ} } .

The map y ↦ x = y^{τ} y^{−1} defines an injection

i : S ↪ C .

This map is not surjective in general. However, if e(E/E^{+}) = 2, then R → S is etale and Sl(2) is smooth and connected. Therefore the map i is surjective, therefore bijective. If e(E/E^{+}) = 1, then E^{+}/F is ramified. Therefore

y^{τ} y^{−1} = id mod π_{E^{+}} ,

whereas x^{τ} = x^{−1} implies x^{2} = id mod π_{E^{+}}, hence only

x = ±id modπ_{E^{+}} .

Indeed if x ∈ C, then also −x ∈ C. But if x = id mod π_{E^{+}} is in C, then one shows easily x = y^{τ} y^{−1} for y ∈ Gl(2, S)^{0} (by recursion mod powers of π_{E^{+}}). But then it is clear, that x ∈ C implies y ∈ S. Therefore C = i(S) ∪ −i(S). We get

#(S) = \frac{1}{2}e(E^{+}/E)#(C) .

So in our counting problem we are reduced to count all x ∈ Sl(2, S) with x = x^{∗} = (x^{τ})^{−1}

and xsx^{−1} = s^{τ}.

\underline{Nonvanishing of C}:

Suppose we are given

s = ( \begin{matrix}\overline{a} & \overline{b}D \\ \overline{b} & \overline{a} \end{matrix} ) .

Consider the set C

C = {x ∈ Sl(2, S) | x^{τ} = x^{−1}; xsx^{−1} = s^{τ} } .

Put \overline{b} = Bπ_{E^{+}}^{ν} and

inv(\overline{b}, D) = (\overline{b}/\overline{b}^{τ})D ∈ S .

Actually this is only welldefined modulo Ann_{S}(\overline{b}), but we later choose some fixed representative. Suppose

C ≠ ∅ .

This implies

inv(\overline{b}, D)^{τ} = inv(\overline{b}, D) mod Ann_{S}(\overline{b}) .

Namely one has (\overline{b}^{τ} /\overline{b})D^{τ} = (x_{1}/x_{1}^{τ})D mod Ann_{S}(\overline{b}) and (x_{1}/x_{1}^{τ})D = (\overline{b}/\overline{b}^{τ})D mod Ann_{S}(\overline{b}).

So we can and will fix some

inv(\overline{b}, D) ∈ R

depending only on s.

\underline{Description of C}:

Then we have already shown, that

C = \{x = ( \begin{matrix}x_{1} & r_{2}\sqrt{A} \\ r_{3}\sqrt{A} & x_{1}^{τ} \end{matrix} ) | det(x) = 1; x_{1}^{τ} = x_{1}(\overline{b}^{τ} /\overline{b}) mod Ann_{S}(\overline{b});

r_{2}\sqrt{A} = inv(\overline{b}, D)r_{3}\sqrt{A} + a for a ∈ Ann_{S}(\overline{b})\} .

where of course

a ∈ Ann_{S}(\overline{b}) ∩ \sqrt{A}R .

We get from \overline{b} = Bπ_{E^{+}}^{ν}

\overline{b}^{τ} /\overline{b} = (B^{τ} /B)(−1)^{eν} .

This gives, modulo Ann_{S}(\overline{b}), the relation (x_{1}/B)^{τ} = (−1)^{eν}(x_{1}/B). But x_{1}/B ∈ S^{∗} and therefore (−1)^{eν} = 1 automatically has to hold. Only the case e = 1, where E^{+}/F is ramified, has to be considered. But then S^{∗} ∩ \sqrt{A}R = ∅. Hence

(x_{1}/B)^{τ} = (x_{1}/B) or x_{1} ∈ B · R .

We get for C

\{x = ( \begin{matrix}r_{1}B(1 + δ) & inv(\overline{b}, D)r_{3}\sqrt{A} + a \\ r_{3}\sqrt{A} & r_{1}B^{τ} (1 + δ)^{τ} \end{matrix} ) | det(x) = 1; a, δ ∈ Ann_{S}(\overline{b}); a^{τ} = −a\} / ≅ ,

where we consider solutions x = (r_{1}, r_{3}, δ, a) and x′ = (r′_{1}, r′_{3}, δ′, a′) equivalent x ≅ x′, if r_{3} = r′_{3}, a = a′ and r_{1}(1 + δ) = r′_{1}(1 + δ′). So we have a description of C in terms of equivalence classes of elements (r_{1}, r_{3}, δ, a) ∈ R^{2} × Ann_{S}(\overline{b})^{2} fullfilling equations

a^{τ} = −a

and

r_{1}^{2}BB^{τ} (1 + δ)(1 + δ)^{τ} − r_{3}^{2}A · inv(\overline{b}, D) − ar_{3}\sqrt{A} = 1 .

We can now choose δ ∈ Ann_{S}(\overline{b}), α ∈ Ann_{S}(\overline{b}\sqrt{A}) ∩ R and r_{3} ∈ R arbitrarily. Put a = \sqrt{A}α. Then there are exactly two solutions r_{1} ∈ R of the equation

r_{1}^{2}BB^{τ} (1 + δ)(1 + δ)^{τ} − r_{3}^{2}A · inv(\overline{b}, D) − αr_{3} = 1 .

This follows, because the coefficient BB^{τ} is a square in R^{∗}. Otherwise C = ∅. Furthermore A·inv(\overline{b}, D) and αA are in the maximal ideal. Extracting square roots from units congruent one is always possible and gives exactly two solutions. The equivalence relations identifies 1 + Ann_{S}(\overline{b}) ∩ R solutions. Therefore

#C = 2#R#Ann_{S}(\overline{b})[(Ann_{S}(\overline{b}) ∩ R\sqrt{A}) : (Ann_{S}(\overline{b}) ∩ R)] .

This is

2#R#Ann_{S}(\overline{b})

in the case where E/E^{+} is ramified and also in the case E/E^{+} unramified, if ord(\overline{b}) = ν is even. However this is satisfied if \overline{b} ≠ 0 and C ≠ ∅.

We get \underline{Lemma 3}: Suppose that the set of solutions S is nonempty. Then the number of solutions #S is either Sl(2, S)/Sl(2, R) if \overline{b} = 0 in S or otherwise

#S = e(E/E^{+})#R#Ann_{S}(\overline{b}) .

\underline{On T \ H-integration}: In this section assume D_{0} ∈ E^{+} and θ ∈ E^{+} to be normalized of order 0 or 1. Consider regular elements

η = ϕ_{θ^{2}D_{0}}(x) ∈ H

with centralizer ϕ_{θ^{2}D_{0}}(T). Later we write H as a disjoint union of double cosets

⋃_{r∈R} ϕ_{θ^{2}D_{0}}(T) · r · K_{0} ,

where the representatives r ∈ Gl(2, E^{+})^{0} are of the form

r ∈ ϕ_{θ^{2}D_{0}}(T) · ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)}ϵ_{0} \end{matrix} ))

for certain j ∈ N and ϵ_{0} ∈ o_{E^{+}}^{∗}. This implies

∫_{H} f(h)dh = ∑_{r}[o_{T} : (T ∩ rK_{0}r^{−1})] ∫_{T} dt ∫_{K_{0}} f(trk)dk ,

hence

∫_{T \H} f(h)dh/dt = ∑_{r}[o_{T} /(T ∩ rK_{0}r^{−1})] ∫_{K_{0}} f(rk)dk .

It allows to calculate the orbital integrals on H by integrations over K_{0} and gives

O_{η}^{G}(1_{K}) = ∑_{i≥0} ∑_{r∈R}[o_{T} /(T ∩ rK_{0}r^{−1})][K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}r^{−1}ηrk)dk .

Observe x = a + b\sqrt{A_{0}} = a + (b/θ)\sqrt{θ^{2}D_{0}}, hence for Θ = θ/π_{E^{+}}^{ord(θ)}

η_{r} := r^{−1}ηr = ϕ_{Θ^{2}D_{0}π_{E_{+}}^{2j} ϵ_{0}^{2}}(x)

because η_{r} is

( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{ord(θ)−j}ϵ_{0}^{−1} \end{matrix} ) ( \begin{matrix}a & bθD_{0} \\ bθ^{−1} & a \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)}ϵ_{0} \end{matrix} ) = ( \begin{matrix}a & _{\frac{b}{Θϵ_{0}π_{E^{+}}^{j}}} (Θϵ_{0}π_{E^{+}}^{j})^{2}D_{0} \\ \frac{b}{Θϵ_{0}π_{E^{+}}^{j}} & a \end{matrix} ) .

Therefore T ∩ rK_{0}r^{−1} ⊂ o_{T} = {x ∈ o_{E}^{∗} | Norm_{E/E^{+}}(x) ∈ o_{F}^{∗} } is the group

\{x = a + b\sqrt{D_{0}} | Norm_{E/E^{+}}(x) ∈ o_{F}^{∗}; a, π_{E^{+}}^{−j}b ∈ o_{E^{+}}\} .

It does not depend on the choice of the (normalized) D_{0} resp. θ.

\underline{Constraints (for fixed r and corresponding j)}: x ∈ T ∩ rK_{0}r^{−1} means x ∈ o_{E}(j) ∩ T.

Furthermore the \underline{index} is

[o_{T} /(T ∩ rK_{0}r^{−1})] = [o_{E}^{∗} : o_{E}(j)^{∗}][(Norm_{E/E^{+}}(o_{E}^{∗}) ∩ o_{F}^{∗}) : (Norm_{E/E^{+}}(o_{E}(j)^{∗}) ∩ o_{F}^{∗})]^{−1} .

Certain constraints have to be satisfied, if the orbital integral

[K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}η_{r}k)dk

should not vanish. Namely k^{−1}η_{r}k ∈ K_{i} implies η_{r} ∈ K_{0}, hence x ∈ T ∩ rK_{0}r^{−1}.

Suppose these r-constraints are satisfied. Then evaluating the orbital integral

[K_{0} : K_{i}] ∫_{K_{0}} 1_{K_{i}}(k^{−1}η_{r}k)dk

amounts to count all cosets kK_{i} ⊂ K_{0}, for which k^{−1}η_{r}k ∈ K_{i}. This is the number of all

cosets y ∈ Gl(2, o_{E^{+}}/π_{F}^{i} o_{E^{+}})^{0}/Gl(2, o_{E^{+}}(i))^{0} in Gl(2, o_{E^{+}}/π_{F}^{i} o_{E^{+}}), such that

y^{−1}η_{r}y ∈ Gl(2, o_{E^{+}}(i))^{0} mod π_{F}^{i}

holds. This number is one for i = 0 and computed in lemma 2 and 3 for i ≥ 1. To apply these lemmas it is enough to show K_{i}/K(π_{F}^{i}) ≅ Gl(2, R), K_{0}/K(π_{F}^{i}) ≅ Gl(2, S)^{0} and Gl(2, o_{E^{+}})/K(π_{F}^{i}) ≅ Gl(2, S) for the normal subgroup K(π_{F}^{i}) of all matrices in Gl(2, o_{E^{+}}), which are congruent 1 modulo π_{R}^{i}. Use that the residue charachteristic is not two,

therefore (1 + π_{F}^{i} o_{E^{+}})^{2} = 1 + π_{F}^{i} o_{E^{+}}.

\underline{Double cosets in H}:

For normalized D_{0} we have

Gl(2, E^{+}) = ⋃_{j≥0} ϕ_{D_{0}}(E^{∗}) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j} \end{matrix} ) Gl(2, o_{E^{+}}) .

This gives a decomposition (with \underline{unique} integers j ≥ 0) for elements h ∈ H

h = ϕ_{θ^{2}D_{0}}(l) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)} \end{matrix} ) k k ∈ Gl(2, o_{E^{+}}), l ∈ E^{∗} .

In order to adjust the data to the group H we are only allowed to change k to k_{0}k for some k_{0} ∈ Gl(2, o_{E^{+}}) with the property

k_{0} ∈ ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)}ϵ_{0} \end{matrix} )^{−1} ϕ_{θ^{2}D_{0}}(E^{∗}) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)}ϵ_{0} \end{matrix} ) ,

which means

k_{0} ∈ o_{E}^{∗}(j) .

Hence k_{0} can be assumed to lie in K_{0}, whenever Norm(o_{E}^{∗}(j))o_{F}^{∗} = o_{E^{+}}^{∗} . This is the case only if j = 0 and E/E^{+} is unramified. In all other cases Norm(o_{E}^{∗}(j))o_{F}^{∗} = (o_{E^{+}}^{∗})^{2} has index two in o_{E^{+}}^{∗}.

On the other hand h ∈ H imposes conditions on det(k). Namely det(h) ∈ F^{∗} implies

(∗) det(k)π_{E^{+}}^{j−ord(θ)} = 1 in (E^{+})^{∗}/(F^{∗} · Norm_{E/E^{+}}(E^{∗})) ≅ H^{1}(F, T) .

In the case, where E/E^{+} is \underline{ramified}, we have chosen π_{E^{+}} = Norm_{E/E^{+}}(π_{E}) = −D_{0} to be in Norm_{E/E^{+}}(E^{∗}). Then (*) implies det(k) ∈ Norm(L^{∗}) ∩ o_{E^{+}}^{∗} = (o_{E^{+}}^{∗})^{2}. Changing k by some k_{0} with determinant in (o_{E^{+}}^{∗})^{2} allows to assume det(k) ∈ o_{F}^{∗}, hence k ∈ K_{0} ⊂ H.

We therefore can write H as a disjoint union of double cosets

⋃_{r∈R} ϕ_{θ^{2}D_{0}}(T) · r · K_{0} , R = N_{0}

with representatives indexed by j ∈ N_{0}

r = r_{j} = ϕ_{θ^{2}D_{0}}(π_{E}^{−j+ord(θ)}) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)} \end{matrix} ) (π_{E^{+}} ∈ F^{∗} ∩ Norm_{E/E^{+}}(E^{∗})) .

Consider the case, where E/E^{+} is \underline{unramified}. Then det(k)π_{E^{+}}^{j−ord(θ)} ∈ F^{∗}Norm_{E/E^{+}}(E^{∗}) implies

j − ord(θ) = 0 mod 2 .

Secondly, to achieve det(k) ∈ o_{F}^{∗}, we are only allowed to change det(k) by elements in the group Norm_{E/E^{+}}(o_{E}^{∗}(j)). This is (o_{F}^{∗})^{2} unless j = 0. Let ϵ_{0} be in o_{E^{+}}^{∗} \ (o_{E^{+}}^{∗})^{2}. Then we get H as a disjoint union of double cosets

⋃_{r∈R} ϕ_{θ^{2}D_{0}}(T) · r · K_{0}

where R ⊂ ((o_{F}^{∗} /(o_{F}^{∗})^{2}) × N ∪ (1 × 0) consists of all elements (ϵ_{0}, j) ∈ (o_{F}^{∗} /(o_{F} ∗)^{2}) × N_{0} with

the property

j = ord(θ) mod 2 .

The representatives are

r = r_{ϵ_{0},j} = ϕ_{θ^{2}D_{0}}(π_{E}^{(−j+ord(θ))/2}ϵ′_{0}) ( \begin{matrix}1 & 0 \\ 0 & π_{E^{+}}^{j−ord(θ)}ϵ_{0} \end{matrix} ) .

Here ϵ′_{0} ∈ o_{E}^{∗} is chosen, such that Norm_{E/E^{+}}(ϵ′_{0}) = ϵ_{0}^{−1} and where ϵ_{0} ∈ o_{E^{+}}^{∗}/(o_{E^{+}}^{∗})^{2}. Furthermore π_{E^{+}} = \sqrt{A_{0}}.

\underline{Summation conditions using ν}:

Our final summation will be indexed by certain integers ν (related to representatives in R in H) and the integers i (coming from H \ G/K double coset). There are several summation conditions controlled among others by the parameters N and and χ = ord(a− a′), which where attached to x ∈ T.

Let us introduce the new parameter 0 ≤ ν = N − j ≤ N. It is introduced, because we have to consider the matrix

s = ( \begin{matrix}a & _{\frac{b}{Θϵ_{0}π_{E^{+}}^{j}}} (Θϵ_{0}π_{E^{+}}^{j})^{2}D_{0} \\ \frac{b}{Θϵ_{0}π_{E^{+}}^{j}} & a \end{matrix} )

as a matrix with coefficients in S = o_{E^{+}}/π_{F}^{i}. With this new variable we get b/π_{E^{+}}^{j} =

Bπ_{E^{+}}^{ν}, b′/π_{E^{+}}^{j} = B′π_{E^{+}}^{ν} with units B, B′ ∈ o_{E^{+}}^{∗}.

Also put θ = Θπ_{E^{+}}^{ord(θ)}. If E/E^{+} is unramified we put θ = π_{E^{+}}.

\underline{Warning (concerning i)}: From now on we make the following change of notation! We replace i by i/e(E^{+}/F). This has the advantage, that π_{F}^{i} becomes π_{E^{+}}^{i} in all formulas. On the other hand we get from now on the extra condition, that i is divisible by 2 in the

case e(E/E^{+}) = 1.

From lemma 2(3) applied to the matrix s above, we get conditions on the summation indices (i, r) ∈ N_{0} × R with r = (ϵ_{0}, j) = (ϵ_{0}, N − ν), which must be satisfied in order to get a nonzero contribution to the orbital integral. These conditions are:

1) 0 ≤ i ≤ χ or a = a′ mod π_{E^{+}}^{i} or (\overline{a} ∈ R)

2) If ν < i (or b/π_{E^{+}}^{j} ≠ 0 in S = o_{E^{+}}/π_{F}^{i}), then (π_{E^{+}}^{ν}BΘ^{−1}ϵ_{0}^{−1})/(π_{E^{+}}^{ν}BΘ^{−1}ϵ_{0}^{−1})^{τ} =

ϵ/ϵ^{τ} mod π_{E^{+}}^{i−ν} has to hold for some ϵ ∈ (o_{E^{+}}^{∗})^{e(E/E^{+})}.

3) If 2N + ord(D_{0}) − ν < i, then i ≤ χ − ν has to hold.

The second condition 2) is the condition ϵ\overline{b} ∈ R of lemma 2(3). Concerning condition 3) observe, that the conditions ϵ\overline{b} ∈ R and ϵ^{−1}\overline{b}D of lemma 2(3) (for some ϵ ∈ (o_{E^{+}}^{∗})^{e(E/E^{+})}) are given in exlicit form by

(∗) (π_{E^{+}}^{ν}BΘ^{−1}ϵ_{0}^{−1})/(π_{E^{+}}^{ν}BΘ^{−1}ϵ_{0}^{−1})^{τ} = ϵ/ϵ^{τ} mod π_{E^{+}}^{i−ν}

and

(∗∗) (π_{E^{+}}^{2N−ν}BΘϵ_{0}D_{0})/(π_{E^{+}}^{2N−ν}BΘϵ_{0}D_{0})^{τ} = ϵ^{τ} /ϵ mod π_{E^{+}}^{i−2N−ord(D_{0})+ν} .

The term on the left and right sides are in o_{F}. They can be combined and in this way simplified to give 3).

To show this, first observe π_{E^{+}}^{2} ∈ F, because either π_{E^{+}} = Norm_{E/E^{+}} \sqrt{D_{0}} = −D_{0} or

π_{E^{+}} = \sqrt{A_{0}}.

Let us multiply (*) and (**). From the way these conditions arise in lemma 2, it is clear that (*) is a congruence condition of order greater or equal to the order of the congruence condition (**). Therefore the multiplied form (***) is still equivalent to (**).

Together with (π_{E^{+}}^{2})^{τ} = (π_{E^{+}}^{2}) this gives

(∗ ∗ ∗) (B^{2}D_{0})/(B^{2}D_{0})^{τ} = 1 mod π_{E^{+}}^{i−2N−ord(D_{0})+ν}

or in other words (multiplication with B^{2}D_{0} shifts the congruence by ord(D_{0}))

(∗ ∗ ∗∗) i − 2N + ν ≤ ord(D_{0}B^{2} − (D_{0}B^{2})^{τ}) .

Condition (***) is only in force, as long as i − 2N − ord(D_{0}) + ν > 0. See condition 3) above. This condition i > 2N + ord(D_{0}) − ν implies i > 0, because N − ν = j ≥ 0. i > 0 implies χ = ord(a − a′) ≥ i > 0 by condition 1), which was obtained above. But χ > 0 implies, that the assumptions of lemma 1 are satisfied. From lemma 1(2) we get

χ = ord(D_{0}(b^{2} − (b′)^{2})). Hence from b = Bπ_{E^{+}}^{N}, b′ = B′π_{E^{+}}^{N} we get

χ = 2N + ord(D_{0}B^{2} − (D_{0}B^{2})^{τ}) .

Therefore the condition (****) can be reformulated into

2N + ord(D_{0}) − ν < i implies i ≤ χ − ν .

This is the condition 3) stated above.

\underline{Résumé}: We now express the results of the chapter on T \H integration in terms, which are suitable for summation. We have to evaluate a sum indexed by 0 ≤ ν ≤ N and 0 ≤ i subject to further conditions. We have gathered everything in the following list. We saw, that the domain of possible values i is divided into two parts subject to the reparametrisation condition, that i is divisible by e(E^{+}/F): \underline{A) Conditions on i} (apart from 0 ≤ i ≤ χ, which is only relevant for χ = 0 ):

\underline{Range of small i}: 0 ≤ i ≤ ν \underline{Large range i}: ν < i ≤ χ − ν

B) Conditions on ν and θ: In the domain of possible values ν we have, in addition to possible restrictions mentioned above, the conditions:

(i) 0 ≤ ν ≤ N

(ii) ν = N + ord(θ) mod 2 ( if e(E/E^{+}) = 1) .

Furthermore in the large range (concerning i) we have contributions only if

(iii) π_{E^{+}}^{ν}(BΘ^{−1}ϵ_{0}^{−1}) ∈ (o_{E^{+}}^{∗})^{e(E/E^{+})} · o_{F} mod π_{E^{+}}^{i}

is satisfied. This forces ν = 0 mod 2 in the large range, if e(E/E^{+}) = 1.

\underline{C) Contributions from the index index(ν)}: The relevant contributions are

1 (ν = N)

\frac{(q + 1)}{2q}q^{N−ν} (ν < N) ,

in the case where E/E^{+} is \underline{unramified} and

q^{N−ν}

in the case, where E/E^{+} is \underline{ramified}. This is shown in the appendix I, 4) resp. II, 5). \underline{D) Contributions from counting solutions sol(ν, i)}:

In the last section we made a reparametrisation of the parameter i, which implies that i has to be divisible by 2, if e(E/E^{+}) = 1. Therefore we set sol(ν, i) = 0 unless this condition is satisfied.

If the condition is satisfied put S = o_{E^{+}}/π_{F}^{i/e(E^{+}/F)} and R = o_{F} /π_{F}^{i/e(E^{+}/F)}. Then sol(ν, i) is given by the number of solutions computed in lemma 3, namely

1 (i = 0)

Alternatively for i ≥ 1

#Sl(2, S)/#Sl(2, R) (i ≤ ν small range)
e(E/E^{+})#R#Ann_{S}(π_{E^{+}}^{ν}) (i > ν large range) .

In principle we now could compute the orbital integrals. We leave this to the reader. We concentrate on the κ-orbital integrals instead.

\underline{The summation (E/E^{+} ramified)}:

For η corresponding to x = t_{1}t_{2}, both t_{1} ∈ L^{∗} and t_{2} ∈ L′ can be assumed to be units. Otherwise our κ-orbital integral is zero. This follows from the r-constraints and the following fact: For the κ- orbital integral we do not get any contribution from the small range of the i-summation in the \underline{ramified} case, because this contribution is stable (does not depend on θ). The large range contributions remain. The whole summation is therefore empty unless χ > 0, i.e. x = x′ mod Π in addition to x, x′ ∈ o_{L}^{∗}. This is true, because χ = 0 implies i = 0, which is in the stable range. This allows to apply the preliminary remarks, made earlier.

Let χ_{0} be the quadratic character of (E^{+})^{∗} attached to E/E^{+}. It is trivial on (o_{E^{+}}^{∗})^{2} with

χ_{0}(θ) = −1. Observe index(ν) = q^{N−ν} and e(E/E^{+}) = 2, #R = q_{0}^{i} and #Ann_{S}(π_{E^{+}}^{ν}) = q^{ν}.

We get from this the contributions q^{N−ν} · (2q_{0}^{i}q^{ν}) = 2q^{N}q_{0}^{i} with q_{0} = q^{1/2}

O_{η}^{κ}(1_{K}) = 2q^{N}χ_{0}(B)l · 1_{o_{L}^{∗}×o_{L′}^{∗}} (t_{1}, t_{2}) ∑_{0≤ν≤N} ∑_{ν<i≤χ−ν} q_{0}^{i} .

The double sum gives

(q_{0} − 1)^{−1} ∑_{0≤ν≤N}(q_{0}^{χ+1−ν} − q_{0}^{ν+1})

= (q_{0} − 1)^{−1}[q_{0}^{χ+1}(q_{0}^{−N−1} − 1)/(q_{0}^{−1} − 1) − q_{0}(q_{0}^{N+1} − 1)/(q_{0} − 1)]

= \frac{q_{0}}{(q_{0} − 1)^{2}} (1 − q_{0}^{N+1})(1 − q_{0}^{χ−N}) = q_{0} ∫_{Gl(2,F)^{2}/(L^{∗})^{2}} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dl .

Hence O_{η}^{κ}(1_{K}) is equal to

2|\frac{x − x^{σ}}{2\sqrt{D_{0}}} |^{−2}χ_{0}(\frac{x^{σ} − x}{2\sqrt{D_{0}}}) · q_{0} ∫_{Gl(2,F)^{2}/(L^{∗})^{2}} 1_{K}(g^{−1}(t_{1}, t_{2})g)dg/dl .

Remember q^{N} = |(x−x^{σ})/2\sqrt{D_{0}}|^{−2} and Bπ_{E^{+}}^{N} = (x−x^{σ})/2\sqrt{D_{0}}. By assumption π_{E^{+}} = −D_{0}

is in Norm_{E/E^{+}}, as E/E^{+} is ramified. The factor 2 in the formula disappears, because in the ramified case [o_{T} : ρ((o_{L}^{∗})^{2})] = 2. Furthermore q_{0} cancels |2\sqrt{D_{0}}|^{2}. Finally χ_{0}(\frac{x−x^{σ}}{2\sqrt{D_{0}}} ) =

χ_{L/F} ((x − x^{σ})(x^{τ} − x^{στ}))χ_{L/F} (2^{−2}D_{0}^{−1}) and χ_{L/F} (2^{−2}D_{0}^{−1}) = χ_{L/F} (−1). Finally remember,

that the change from ϕ to ϕ̌ contributes a factor χ_{L/F} (−A_{0}). If we multiply together, the theorem follows in the case, where E/E^{+} is ramified.

\underline{The summation (E/E^{+} unramified)}:

First of all we get the support condition, that t_{1}, t_{2} have to be units. To compute the κ-orbital integral we concentrate on the case χ > 0. The case χ = 0 is excluded by the parity rule, χ is always odd. From the résumé we get the following contributions:

1) \underline{Small term range}: Its contribution will be

SRC = ∑_{0≤ν≤N} ∑_{0≤i≤ν}(−1)^{N−ν}#(ϵ_{0} − reprs.) · index(ν) · sol(i)

= ∑_{0≤i≤N} sol(i)( ∑_{i≤ν<N}(−1)^{N−ν} · 2 · \frac{(q + 1)}{2q}q^{N−ν} + ∑_{ν=N} 1) .

The representatives ϵ_{0} appearing in R give the factors 2 in the first inner sum! The second inner sum has only one term! The computation of the inner sum gives

SRC = ∑_{0≤i≤N} sol(i)(−q)^{N−i} .

The residue fields of E^{+} and F have the same cardinality q. Therefore for even i the

number sol(i) = #Sl(2, S)/#Sl(2, R) becomes q^{dim(Sl(2))i/2} = q^{3i/2}. Therefore sol(i) = q^{3i/2}

or 0 depending on the parity of i (see resume part D)). Thus

SRC = (−q)^{N} ∑_{0≤j≤N/2} q^{j} = \frac{(−q)^{N}}{(q − 1)^{2}} [(q − 1)q^{1+[N/2]} − (q − 1)] .

Here [x] is the largest integer n ≤ x.

2) \underline{Large range contribution}: We get

LRC = ∑_{ν<i≤χ−ν} ∑_{0≤ν≤N}(−1)^{N−ν} · #(ϵ_{0} − reprs.) · index(ν) · sol(ν, i) ,

where summation is over all even integers ν. See Resume B(iii), in the case e(E/E^{+}) = 1 for the large range. The number sol(ν, i) = 0 unless i are even. Furthermore for even i

we have e(E/E^{+}) = 1, #R = q^{i/2} and #Ann_{S}(π_{E^{+}}^{ν}) = q^{ν}. Therefore sol(ν, i) = q^{i/2+ν} for

even i. Hence

LRC = ∑_{ν<i≤χ−ν,i even}(  ∑_{0≤ν<N,ν even}(−1)^{N} · 2 · \frac{q + 1}{2q}q^{N−ν} · q^{i/2+ν})

+( ∑_{N<i≤χ−N,i,N even}(−1)^{N} · 1 · q^{i/2+N} · 1) .

Summation over i and then ν gives

\frac{(−q)^{N}}{(q − 1)^{2}} ((q + 1)q^{\frac{χ+1}{2}} − (q + 1)q^{\frac{χ−N}{2}} − (q + 1)q^{\frac{N+1}{2}} + (q + 1))

in the case N \underline{odd} and

\frac{(−q)^{N}}{(q − 1)^{2}} ((q + 1)q^{\frac{χ+1}{2}} − 2q^{\frac{χ−N+1}{2}} − (q^{2} + 1)q^{\frac{N}{2}} + (q + 1))

in case N is \underline{even}.

3) \underline{Total contribution}: Adding together the contributions from 1),2) gives

SRC + LRC = \frac{(−q)^{N}}{(q − 1)^{2}} (−2 + (q + 1)q^{\frac{ev}{2}} )(−1 + qq^{−1/2}q^{\frac{odd}{2}} ) ,

where ev is the even number among N, χ − N and odd is the odd one (parity rule). But

|D_{0}| = 1, |A_{0}| = q^{−1/2}, q = q_{0}, (−1)^{N} = (−1)^{ord(π_{E^{+}}^{N} B)} = χ_{0}((x − x^{σ})/2\sqrt{D_{0}}) = χ_{L/F} ((x −

x^{σ})(x^{στ} − x^{τ})). In the present case the first of the two Gl(2, F)-orbital integrals (for M) corresponds to the unramified extension L/F, the second to the ramified extension L′/F. Finally take into account the factor χ_{L/F} (−A_{0}), which comes from our change from ϕ to ϕ̌. If we multiply together, the theorem is also proved in the unramified case.

\underline{References}:

[H] Hales T.C., Shalika germs on GSp(4), asterisque 171-172, p.195 - 256 (1989)

[LS] Langlands R.P. -Shelstad D., On the definition of transfer factors, Math. Ann. 278, p. 219-271 (1987)

[S] Schröder M., Thesis (university of Mannheim), 1993

[W] Weissauer R., A special case of the fundamental lemma I ( The case GSp(4,F)), Mannheim 1993 \underline{Appendix I on orders (E/E^{+} ramified)}: Let the situation be as above. Then the orders o_{E}(j) = o_{E^{+}} + π_{E^{+}}^{j}o_{E} ⊂ o_{E} (where j > 0 by assumption) have the following properties:

1) Norm(o_{E}(n)^{∗}) = (o_{E^{+}}^{∗})^{2} This is clear from (o_{E^{+}}^{∗})^{2} ⊂ Norm(o_{E}(n)^{∗}) ⊂ Norm(o_{E}^{∗}) ⊂ (o_{E^{+}}^{∗})^{2}.

2) [o_{E}^{∗}(n) : (1 + π_{E^{+}}^{n}o_{E})] = [o_{E^{+}}^{∗} : (o_{E^{+}}^{∗} ∩ (1 + π_{E^{+}}^{n}o_{E})] = [o_{E^{+}}^{∗} : (1 + π_{E^{+}}^{n}o_{E^{+}})] = (q − 1)q^{n−1} 3) [o_{E}^{∗} : o_{E}^{∗}(n)] = [o_{E}^{∗} : (1 + π_{E^{+}}^{n}o_{E})]/[o_{E}^{∗}(n) : (1 + π_{E^{+}}^{n}o_{E})] = (q − 1)q^{2n−1}/(q − 1)q^{n−1} = q^{n} 4) The index is [o_{T} : (T ∩ rK_{0}r^{−1})] = [o_{L}^{∗} : o_{L}^{∗}(j)] = q^{j} because (o_{E}^{∗})^{(σ+1)(τ−1)} =

(o_{E}(j)^{∗})^{(σ+1)(τ−1)} follows from 1). Finally for S = o_{E^{+}}/π_{E^{+}}^{i}o_{E^{+}} and i > 0

5) #(Sl(2, S)) = (q − 1)(q + 1)q^{3i−2}

\underline{Appendix II on orders (E/E^{+} unramified)}: Let the situation be as above. Let o_{E}(j) = o_{E^{+}} + π_{E^{+}}^{j}o_{E} ⊂ o_{E} for j > 0.

1) Norm(o_{E}(j)^{∗}) = (o_{E^{+}}^{∗})^{2} if j > 0 and Norm(o_{E}^{∗}) = o_{E^{+}}^{∗} This is clear from (o_{E^{+}}^{∗})^{2} ⊂ Norm(o_{E}^{∗}(j)) ⊂ Norm(o_{E}^{∗}) = (o_{E^{+}}^{∗}) and [o_{E^{+}}^{∗} : (o_{E^{+}}^{∗})^{2}] = 2 and o_{E}^{∗}(j) ⊂ o_{E^{+}}^{∗}(1 + π_{E}o_{E}).

2) For n > 0 we have [o_{E}^{∗}(n) : (1 + π_{E^{+}}^{n}o_{E})] = [o_{E^{+}}^{∗} : (o_{E^{+}}^{∗} ∩ (1 + π_{E^{+}}^{n}o_{E})] = [o_{E^{+}}^{∗} : (1 + π_{E^{+}}^{n}o_{E^{+}})]= (q − 1)q^{n−1}

3) For n > 0 we have [o_{E}^{∗} : o_{E}^{∗}(n)] = [o_{E}^{∗} : (1 + π_{E^{+}}^{n}o_{E})]/[o_{E}^{∗}(n) : (1 + π_{E^{+}}^{n}o_{E})] =

(q^{2} − 1)q^{2(n−1)}/(q − 1)q^{n−1} = (q + 1)q^{n−1}.

Using 1) and 3) we get for the index:

5) [o_{T} : (T ∩ rK_{0}r^{−1})] is 1 or \frac{1}{2}(q + 1)q^{j−1}. This follows because Norm_{E/E^{+}}(o_{E}^{∗}) ∩ o_{F}^{∗} =

Norm_{E/E^{+}}(o_{E}(j)^{∗}) ∩ o_{F}^{∗} for j = 0 and has index two for j > 0. Namely in this case (o_{E^{+}}^{∗})^{2} ∩ o_{F}^{∗} = (o_{F}^{∗})^{2} has index two in o_{E^{+}}^{∗} ∩ o_{F}^{∗} = o_{F}^{∗}.

This value is

1 , (j = 0)

\frac{1}{2}(q + 1)q^{j−1} , (j > 0) .

\underline{Appendix III}:

Let us descibe the admissable embedding T_{M} ↪ T_{G}. We have the following chain of embeddings:

M = Gl(2, F)^{2}/F^{∗} ↩ T_{M} ↩ T = (L^{∗}) × (L′)^{∗}/F^{∗} →^{ι} E^{0} ⊂ E

E^{0} →^{ϕ\sqrt{D_{0}}} Gl(2, E^{+})^{0} → Gl(2, E^{+})^{0} → H → H ⊂ G = GSp(4, F) .

The exponent ^{0} indicates, that the relative norm from E to E^{+} respectively the determinant has values in F. The maps involved are, from left to right: First the embedding defined by a regular representation identifying Gl(2, F) and Gl(L/F), which is unique up to conjugation. Then the map ι defined by

ι : (t_{1}, t_{2})/ ∼ ↦ x = t_{1}t_{2}

which has image in the set E^{0}. Thirdly the map

ϕ\sqrt{D_{0}} : x ↦ s = ( \begin{matrix}a & D_{0}b \\ b & a \end{matrix} )

where the coefficients a, b ∈ E^{+} are determined by writing x = t_{1}t_{2} = a + b\sqrt{D_{0}}, where L = F (\sqrt{D_{0}}) and all elements x, t_{1}, t_{2} are viewed as elements of the field E = LL′ of degree 4 over F. Finally the composite ϕ̌ of the following maps from Gl(2, E^{+})^{0} to GSp(4, F). We get ϕ̌ by first applying the outer (!) automorphism

Ad ( \begin{matrix}1 & 0 \\ 0 & \sqrt{A_{0}} \end{matrix} ) ,

then the morphism

ϕ : Gl(2, E^{+})^{0} → H

and then the inner automorphism

Adϕ(( \begin{matrix}1 & 0 \\ −μ & μ \end{matrix} )) ,

and finally the inclusion of H into GSp(4, F). Of course, the second and third part of the last map can be reversed in their order by making first a conjugating with ( \begin{matrix}1 & 0 \\ −μ & μ \end{matrix} ) and then appying ϕ. In any case we obtain as result the element η ∈ GSp(4, F).

The torus T_{G} in GSp(4, F) is defined by

T_{G} = ϕ̌ ∘ ϕ\sqrt{D_{0}} ∘ ι(T_{M}) .

One should also keep in mind the following notations. The field E is the composite of the three quadratic extension fields E^{+} = F (\sqrt{A_{0}}), L = F (\sqrt{D_{0}}) and L′ = F (\sqrt{A_{0}D_{0}}) of F. They are the fixed fields of the automorphisms σ, τ, σ ∘ τ respectively. L/F is ramified or unramified depending on the corresponding behaviour of E/E^{+}.

A special Case of the Fundamental Lemma III

The Langlands-Shelstad Transfer Factor for GSp(4, F)

R.Weissauer Mannheim, January 1994 In this note we verify the global conjecture for the transfer factors studied in the previous part [W] of these notes. This global conjecture holds for the transfer factors defined by Langlands and Shelstad [L-S], (6.4). Therefore it is enough to show, that they agree with the ones used in [W]. To know this will be enough for our applications, as shown in part IV.

\underline{An admissable homomorphism}:

Let the notations and assumptions be as in [W]. Especially F will be a local field of characteristic zero and GSp(4, F) the group of symplectic similitudes of four variables. For the group GSp(4, F) there is exists, up to equivalence, a unique nontrivial elliptic endoscopic group M = Gl(2, F)^{2}/F^{∗}. The subgroup F^{∗}, which is divided out, is embedded in the antidiagonal way as the set of all {(t · id, t^{−1} · id)}. The elliptic tori of M are of the form T_{M} ≅ (L_{1}^{∗} × L_{2}^{∗})/F^{∗} for quadratic extension fields L_{1}, L_{2} of F. The tori with L_{1} = L_{2} will be called elliptic tori of the first type, the other ones of the second type.

In the following we consider tori of the first type. Let L be a quadratic extension field of F with a nontrivial field automorphism σ fixing F. The corresponding torus T_{M} has as F-rational points the elements (t_{1}, t_{2}) ∈ (L^{∗})^{2}/F^{∗}. It can be embedded into M by the regular embedding. Any two such embeddings are conjugate under M(F). Of course, we have to identify the elements (t_{1}, t_{2}) and (t_{1}t, t_{2}t^{−1}). However

x = t_{1}^{σ}t_{2}^{σ} ∈ L^{∗}

and

x′ = t_{1}t_{2}^{σ} ∈ L^{∗}

are well defined. Note, that this notation slightly differs from the one used in [W].

\underline{The split tori T_{M}^{s} and their dual T̂_{M}^{s}}:

Over the algebraic closure, the torus T_{M} is conjugate to the diagonal torus T_{M}^{s}, which consists of all elements

( \begin{matrix}t_{1} & ∗ \\ 0 & t_{1}^{σ} \end{matrix} ) × ( \begin{matrix}t_{2} & 0 \\ ∗ & t_{2}^{σ} \end{matrix} ) mod (t · id, t^{−1} · id) .

The stars beside the diagonal indicate, that we have fixed a reference Borel group containing the split torus T_{M}^{s} over \overline{F}. The positive roots are R_{M}^{+} = \{t_{1}/t_{2}^{σ}, t_{2}^{σ}/t_{2}\} = \{x^{σ}/x′, x′/x\}. We get root-data for M

X^{∗} = X^{∗}(T_{M}^{s}) = {(a, b, c, d) ∈ ZZ^{2} ⊕ ZZ^{2} | a + b = c + d} X_{∗} = X_{∗}(T_{M}^{s}) = ZZ^{2} ⊕ ZZ^{2}/(1, 1, −1, −1)ZZ Δ^{+} = Δ_{+} = {(1, −1, 0, 0), (0, 0, −1, 1)} .

The Galois group acts via σ and permutes the two coordinates of each ZZ^{2} block.

The dual torus T̂_{M}^{s} is the complex torus X^{∗}(T_{M}^{s})⊗C^{∗} = {(z_{1}, z_{2}, z_{3}, z_{4}) ∈ (C^{∗})^{4} | z_{1}z_{2} = z_{3}z_{4}} with σ acting by permuting z_{1} and z_{2} resp. z_{3} and z_{4}. The induced positive coroots are

t ↦ (t, t^{−1}, 1, 1) and t ↦ (1, 1, t^{−1}, t).

\underline{The fixed reference tori T̂_{M} = T̂_{G}}:

Next consider the dual M̂ embedded as the subgroup of all elements

( \begin{matrix}∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \\ ∗ & 0 & ∗ & 0 \\ 0 & ∗ & 0 & ∗ \end{matrix} )

into the dual group Ĝ = GSp(4, C) of G = GSp(4, F). Notation is here as in [W]. We specify the diagonal tori T̂_{M} = T̂_{G} and the Borel groups B̂_{M} = B̂_{G} ∩ M̂, where

B̂_{G} = \{( \begin{matrix}∗ & 0 & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \\ 0 & 0 & 0 & ∗ \end{matrix} )\} .

We can identify T̂_{M}^{s} and T̂_{M} via the isomorphism

(z_{1}, z_{2}, z_{3}, z_{4}) ∈ T̂_{M}^{s} ↦ diag(z_{1}, z_{4}, z_{2}, z_{3}) ∈ T̂_{M} = T̂_{G} .

This map preserves the positive coroots, because the positive coroots for T̂_{M} are given by t ↦ (t, 1, t^{−1}, 1) and t ↦ (1, t, 1, t^{−1}). The same holds also for the positive roots.

\underline{The tori T_{G}^{s} and its dual T̂_{G}^{s}}:

Next we may consider the split diagonal torus T_{G}^{s} in G over the algebraic closure \overline{F}. We choose the Borel group containing T_{G}^{s} consisting of the matrices of the form above (now with entries in \overline{F} instead of C). The diagonal entries will be labeled

diag(x_{1}, x′_{1}, x_{2}, x′_{2}) ∈ G(\overline{F}) .

Observe x_{1}x_{2} = x′_{1}x′_{2}. We have identifications

X^{∗} = X^{∗}(T_{G}^{s}) = {(m_{1}, m_{2}, m′_{1}, m′_{2}) ∈ (ZZ^{2} ⊕ ZZ^{2})/(1, 1, −1, −1)ZZ} X_{∗} = X_{∗}(T_{G}^{s}) = {(m_{1}, m_{2}, m′_{1}, m′_{2}) ∈ ZZ^{2} ⊕ ZZ^{2} | m_{1} + m_{2} = m′_{1} + m′_{2}} X^{∗} ⊃ Φ^{+} = {(−1, 0, 1, 0), (1, −1, 0, 0), (0, 0, 1, −1), (1, 0, 0, −1)} .

\underline{Warning}: Note the reordering (!). Namely (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in

X^{∗}(T_{G}^{s}) corresponds to the characters, which map diag(x_{1}, x′_{1}, x_{2}, x′_{2}) to x_{1} resp. x_{2} resp. x′_{1} resp. x′_{2}. This is also indicated by the notation.

The first two positive roots in the set Φ^{+} will be called α_{1} and α_{2}. They form a basis

Δ = {α_{1}, α_{2}}.

\underline{The tori T_{G} and their duals T̂_{G}}:

Note that the reordering has the following reason. Consider the torus T_{G} of all elements

η = ( \begin{matrix}a & 0 & bD_{0} & 0 \\ 0 & a′ & 0 & b′D_{0} \\ b & 0 & a & 0 \\ 0 & b′ & 0 & a′ \end{matrix} )

in G(F), where L = F (\sqrt{D_{0}}) and D_{0} is normalized. Put

h = ( \begin{matrix}−\sqrt{D_{0}} & 0 & −\sqrt{D_{0}} & 0 \\ 0 & −\sqrt{D_{0}} & 0 & −\sqrt{D_{0}} \\ −1 & 0 & 1 & 0 \\ 0 & −1 & 0 & 1 \end{matrix} ) .

This is an element in G(\overline{F}). It is not in G_{der}(\overline{F}). But we can write it in the form h = h_{der} ·z for a suitable element in the center Z(G)(\overline{F}). We have h^{−1}T_{G}h = T_{G}^{s}, similar for h_{der}. We get in both cases

h^{−1}ηh = diag(x, x′, x^{σ}, (x′)^{σ}) .

Here x = a + b\sqrt{D_{0}} and x′ = a′ + b′\sqrt{D_{0}}.

The Galois group acts on X^{∗}(T_{G}) via its projection on Gal(L/F). We get a new action on X^{∗}(T_{G}^{s}) via transport of structure (coming from the action on X^{∗}(T_{G}) and the diagonalisation). It permutes the entries of the first and second ZZ^{2}-blocks and explains our way of writing for X^{∗}(T_{G}^{s}) in the last section. Of course, one has a corresponding new action on

X_{∗} = X_{∗}(T_{G}^{s}) = {(m_{1}, m_{2}, m′_{1}, m′_{2}) | m_{1} + m_{2} = m′_{1} + m′_{2}}
(Φˇ )^{+} = {(−1, 1, 1, −1), (1, −1, 0, 0), (0, 0, 1, −1), (1, −1, 1, −1)}

The cobase Δˇ is the set {α̌_{1}, α̌_{2}} defined by the first two elements of the set (Φˇ )^{+}.

In the following we will identify T̂_{G} with T̂_{G}^{s} as groups. Only the Galois action will differ, as indicated above. The dual tori T̂_{G}^{s} and T̂_{G} are therefore in both cases defined by X^{∗} ⊗ C^{∗}, or explicitely by

{(w_{1}, w_{2}, w′_{1}, w′_{2}) ∈ (C^{∗})^{2} × (C^{∗})^{2}}/{(t, t, t^{−1}, t^{−1})} .

In the first case the Galois action is trivial, in the second case (viewed as the dual torus of T_{G}) the Galois action factors over Gal(L/F) and σ permutes the coordinates in the two (C^{∗})^{2}-blocks.

\underline{The isomorphism between T̂_{G} and T̂_{G}^{s} resp. T̂_{G}}:

In order to define an admissable embedding

T_{M} ≅ T_{G}

in the sense of [L-S] (1.3), one has to give certain isomorphisms

T̂_{G} ≅ T̂_{G} = T̂_{M} ≅ T̂_{M} .

For this purpose one identifies T̂_{M} with T̂_{M}^{s} (twisted Galois action) and T̂_{G} with T̂_{G}^{s} (twisted Galois action). The isomorphism T̂_{M}^{s} ≅ T̂_{M} was already specified above. It was defined by

(z_{1}, z_{2}, z_{3}, z_{4}) ∈ T̂_{M}^{s} ↦ diag(z_{1}, z_{4}, z_{2}, z_{3}) ∈ T̂_{M} = T̂_{G} .

It remains to give the corresponding isomorphism T̂_{G}^{s} → T̂_{G}. It will map the element

(w_{1}, w_{2}, w′_{1}, w′_{2}) mod (t, t, t^{−1}, t^{−1}) ∈ T̂_{G}^{s}

to the diagonal matrix

diag(w_{2}w′_{1}, w_{1}w′_{1}, w_{1}w′_{2}, w_{2}w′_{2}) ∈ T̂_{G} .

In other words z_{1} = w_{2}w′_{1}, z_{2} = w_{1}w′_{2}, z_{3} = w_{2}w′_{2} and z_{4} = w_{1}w′_{1}, if we identify T̂_{G} and T̂_{M}. This map is well defined and injective, therefore an isomorphism. This isomorphism maps the basis of positive roots given by the characters (−1, 1, 1, −1), (1, −1, 0, 0) or w_{2}w′_{1}/w_{1}w′_{2}, w_{1}/w_{2}, to the basis of positive roots z_{1}/z_{2}, z_{4}/z_{1} of T̂_{G}. Similar, the basis of coroots t ↦ (t^{−1}, 1, t, 1) resp. t ↦ (t, t^{−1}, 1, 1) of T̂_{G}^{s} maps to the basis of coroots

t ↦ diag(t, 1, t^{−1}, 1) resp. t ↦ diag(t^{−1}, t, t, t^{−1}) of T̂_{G}.

The induced map T̂_{G}^{s} → T̂_{G} = T̂_{M} → T̂_{M}^{s} maps (w_{1}, w_{2}, w′_{1}, w′_{2}) to diag(z_{1}, z_{2}, z_{3}, z_{4}) = diag(w_{2}w′_{1}, w_{1}w′_{2}, w_{2}w′_{2}, w_{1}w′_{1}).

\underline{Lemma 1}: Its dual map ϕ : T_{M}^{s} → T_{G}^{s} is

ϕ((t_{1}, t_{1}^{σ}, t_{2}, t_{2}^{σ}) mod (t, t, t^{−1}, t^{−1})) = (x, x^{σ}, x′, (x′)^{σ}) = diag(x, x′, x^{σ}, (x′)^{σ}) ,

where

x = t_{1}^{σ}t_{2}^{σ} , x′ = t_{1}t_{2}^{σ} .

ϕ is an admissable isomorphism of T_{M} onto T_{G}.

\underline{Proof}: From the construction it is clear, that ϕ is an admissable homomorphism. It is therefore enough to check, that ϕ^{∗} induces the correct map of T̂_{G}^{s} → T̂_{M}^{s}. But ϕ^{∗} :

X^{∗}(T_{G}) → X^{∗}(T_{M}) maps (a, b, c, d) mod ZZ(1, 1, −1, −1) to (b + c, a + d, b + d, a + c). Therefore (1, 0, 0, 0) is mapped to (0, 1, 0, 1) and (0, 0, 1, 0) to (1, 0, 0, 1). This means x ↦ t_{1}^{σ}t_{2}^{σ} and x′ ↦ t_{1}t_{2}^{σ}.

\underline{Resume}: We considered tori T_{G}, T_{G}^{s}, T_{M}, T_{M}^{s} and their duals and the reference tori T̂_{M} = T̂_{G}. The torus T_{G} ⊂ GSp(4, F) is identified with the tori T_{M} ⊂ M(F) by an admissable isomorphism ϕ. Over the algebraic closure we can diagonalize both tori and obtain T_{G}^{s} and T_{M}^{s}. For instance we have

h^{−1} ( \begin{matrix}a & 0 & bD_{0} & 0 \\ 0 & a′ & 0 & b′D_{0} \\ b & 0 & a & 0 \\ 0 & b′ & 0 & a′ \end{matrix} ) h = ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & x′ & 0 & 0 \\ 0 & 0 & x^{σ} & 0 \\ 0 & 0 & 0 & (x′)^{σ} \end{matrix} ) ,

where x = a + b\sqrt{D} and x′ = a′ + b′\sqrt{D} and h was defined above. with x = t_{1}^{σ}t_{2}^{σ}, x′ = t_{1}t_{2}^{σ}. The admissable isomorphism is given ’in diagonalized form’ by

( \begin{matrix}t_{1} & 0 \\ 0 & t_{1}^{σ} \end{matrix} ) × ( \begin{matrix}t_{2} & 0 \\ 0 & t_{2}^{σ} \end{matrix} ) mod (t · id, t^{−1}id) ↦ ( \begin{matrix}x & 0 & ∗ & ∗ \\ ∗ & x′ & ∗ & ∗ \\ 0 & 0 & x^{σ} & ∗ \\ 0 & 0 & 0 & (x′)^{σ} \end{matrix} ) .

The admissable isomorphism between the tori T_{M} and T_{G} is defined by the composition with the diagonalisation isomorphisms (i.e. g ↦ hgh^{−1} maps (t_{1}, t_{2}) to an element η ∈ T_{G} = hT_{G}^{s}h^{−1}). To be precise, we also have to fix some embedding of T_{M} into M and some conjugation isomorphism from T_{M} to the split torus in M defined over \overline{F}. This allows to identify the element (t_{1}, t_{2}) ∈ T_{M} with the diagonal matrix in T_{M}^{s}(\overline{F}) considered above.

Statement of Result

With these data fixed, we will now determine the transfer factor of Langlands-Shelstad. It is a product of several terms, listed below. Let denote χ_{L/F} the quadratic character of the local field F^{∗}, attached to the extension L/F by classfield theory. Then

\underline{Lemma 2}: The transfer factor of Langlands and Shelstad [L-S] (3.7), in the quasisplit situation descibed above, is given as the product of the factors

Δ_{I}(η, t_{1}, t_{2}) = χ_{L/F} (D_{0}) = χ_{L/F} (−1)

Δ_{II}(η, t_{1}, t_{2}) = χ_{0}(\frac{x/x^{σ} − 1}{\sqrt{D_{0}}})χ_{0}(\frac{x′/(x′)^{σ} − 1}{\sqrt{D_{0}}})

where χ_{0} is a character on L^{∗} inducing on the subgroup F^{∗} the quadratic character χ_{L/F}.

Δ_{III_{1}}(η, t_{1}, t_{2}) = 1

trivially under our assumptions, which imply γ = γ_{G} in the sense of [L-S],(3.4) .

Δ_{III_{2}}(η, t_{1}, t_{2}) = χ_{0}(x^{σ}/x′)
Δ_{IV} (η, t_{1}, t_{2}) = |xx′|^{−1}|(x^{σ} − x)((x′)^{σ} − x′)| .

This gives the complete factor

Δ(η, t_{1}, t_{2}) = (χ_{L/F} |.|)((x − x^{σ})(x′ − (x′)^{σ}))|xx′|^{−1} .

For the last statement observe |x| = |x′|.

\underline{Remark}: In [W] we used a different assignment x = t_{1}t_{2} and x′ = t_{1}t_{2}^{σ}. However, this change is irrelevant. If we replace x by x^{σ}, the factor of Langlands-Shelstad picks up a sign χ_{L/F} (−1). But this sign describes exactly the change of η in its stable conjugacy under the homomorphism κ (or Δ_{III_{1}}), which is involved in the definition of the κ-orbital integral. The lemma 2 above therefore shows, by comparison with [W1]

\underline{Corollary}: The fundamental lemma for the tori T_{M} is true for the unit element of the Hecke algebra, using the Langlands-Shelstad transfer factor.

The a-data and χ-data

In this section we define the a-data and χ-data ([L-S], (2.2) and (2.5)).

1) The positive roots R_{G}^{+} of T_{G} are the characters, which assign to diag(x, x′, x^{σ}, x^{′σ}) the values x′/x, x/x^{σ}, x′/(x′)^{σ}, x/(x′)^{σ}. The positive roots, which do not occur in M, are x/x^{σ} and x′/(x′)^{σ}. Remember, the Galois group acts via the quotient Gal(L/F). The action of σ turns every root into its negative, because xx^{σ} = x′(x′)^{σ}.

This shows ([L-S], (3.6))

Δ_{IV} (η; (t_{1}, t_{2})) = |(1 − x′/x)(1 − x/x^{σ})(1 − x′/(x′)^{σ})(1 − x/(x′)^{σ})|/|(1 − t_{1}/t_{1}^{σ})(1 − t_{2}^{σ}/t_{2})|

= |(1 − x/x^{σ})(1 − x′/(x′)^{σ})|
= |(x − x^{σ})(x′ − (x′)^{σ})|/|xx′| .

2) Now define a-data for the root system R_{G} by

\begin{aligned}a_{λ} & = \sqrt{D_{0}}  λ ∈ R_{G}^{+} \\ a_{λ} & = −\sqrt{D_{0}}  λ ∈ R_{G}^{−} \end{aligned} .

These numbers satisfy a_{λ} ∈ \overline{F}^{∗} and σ(a_{λ}) = a_{σ(λ)} and a_{−λ} = −a_{λ}, as required in the definition [L-S], (2.2)

3) Similar we get the χ-data. Let us fix a quasi character χ_{0} : L^{∗} → C^{∗}, such that the restriction of χ_{0} from L^{∗} to F^{∗} is the quadratic character χ_{L/F} of F^{∗}, attached to L/F via class field theory. Then we get as χ-data

\begin{aligned}χ_{λ} & = χ_{0}  λ ∈ R_{G}^{+} \\ χ_{λ} & = χ_{0}^{−1}  λ ∈ R_{G}^{−} \end{aligned} .

Namely χ_{σ(λ)} = χ_{λ} ∘ σ^{−1} and χ_{−λ} = χ_{λ}^{−1} hold because of χ_{0} ∘ σ^{−1} = χ_{0}^{−1}. These are the required conditions. See [L-S], (2.5). In the following χ_{0} will be also viewed as a character of the Weil group

W_{+} = W_{L/F_{+}} = W_{L/L} ≅ L^{∗} .

4) Let me remind the reader of some of the notations from [L-S], (2.5) . F_{+λ} is the fixed

field of Γ_{+λ} = {σ ∈ Γ | σ(λ) = λ} and F_{±λ} is the fixed field of Γ_{±λ} = {σ ∈ Γ | σ(λ) = ±λ}.

Γ is the absolute Galois group of F. In our case F_{+} = L and F_{±} = L holds independent from λ, and the index is always two.

5) The positive roots α of G, which do not occur in M are given by the characters x/x^{σ} and x′/(x′)^{σ}, in the sense above. According to [L-S], (3.3) the product of the corresponding terms χ_{α}(\frac{α(η)−1}{a_{α}}) gives the factor Δ_{II}(η; (t_{1}, t_{2})). Hence

Δ_{II}(η; (t_{1}, t_{2})) = χ_{0}(D_{0}^{−1}(x/x^{σ} − 1)(x′/(x′)^{σ} − 1)) .

The element ω_{T} (σ)

This section refers to [L-S], (2.1) and (2.3).

Consider the split torus T_{G}^{s} and the Borel group B_{G}^{s} above. Then conjugation with the matrix h, defined above, gave

h^{−1}(T_{G}^{s}, B_{G}^{s})h = (T_{G}, B_{G}) ,

with a Borel group B_{G} defined over \overline{F}. We get

(h^{−1}h^{σ})^{−1}(T_{G}^{s}, B_{G}^{s})(h^{−1}h^{σ}) = (T_{G}^{s}, B_{G}^{s}) .

The element ω_{T} (σ) is defined by

ω_{T} (σ) = Int(h^{−1}h^{σ}) .

It is in the normalizer of T_{G}^{s}. In our case

h^{−1}h^{σ} = h^{−σ}h = ( \begin{matrix}0 & −E \\ −E & 0 \end{matrix} ) .

Let w_{max} denote the element of maximal length in the Weyl group of the torus T_{G}^{s}. Then the induced element in the Weyl group is: ω_{T} (ρ) ≡ 1 for ρ ∈ Γ_{L} and ω_{T} (ρ) ≡ w_{max} for

ρ ∈ σΓ_{L}.

Lifting of Weyl group

This section refers to [L-S], (2.1) and (2.3).

The simple roots of (T_{G}^{s}, B_{G}^{s}) are α_{1} = 2e_{1} and α_{2} = e_{2} − e_{1}. We get the embedded root subgroups

Sl(2)_{α_{1}} ∋ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}a & 0 & b & 0 \\ 0 & 1 & 0 & 0 \\ c & 0 & d & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

and

Sl(2)_{α_{2}} ∋ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}d & −c & 0 & 0 \\ −b & a & 0 & 0 \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{matrix} ) ,

which map ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) to the Borel group B_{G}^{s}. The standard element of the Weyl group of Sl(2) therefore maps to

n(σ_{α_{1}}) = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

and

n(σ_{α_{2}}) = ( \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) .

The element w_{max} in the Weyl group of GSp(4) can be written as product of simple reflections of lenght four w_{max} = σ_{α_{2}}σ_{α_{1}}σ_{α_{2}}σ_{α_{1}}. Hence the product n(σ_{α_{2}})n(σ_{α_{1}})n(σ_{α_{2}})n(σ_{α_{1}}) is

( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix} )^{2} = ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} ) .

For θ_{1}, θ_{2} in the subgroup of the Weyl group Ω generated by 1 and w_{max}, this lifting to the normalizer defines a cocycle t(θ_{1}, θ_{2}). It has value 1 except in the case θ_{1} = θ_{2} = w_{max}, where it has value −1.

Coroot modification

In this section we compute the class λ_{T} and λ_{T_{sc}} defined in [L-S], (2.3). If s ∈ Z(M̂)^{Γ}, which is satisfied in our case, one can work with λ_{T} instead of λ_{T_{sc}} to define the factor Δ_{I} ([L-S], (3.2). This is conveniant and will be explained at the end of the section.

1) The coroots of T_{G}^{s} are given by the maps, which attach to t ∈ F^{∗} the matrices

diag(t^{−1}, t, t, t^{−1}), diag(t, 1, t^{−1}, 1), diag(1, t, 1, t^{−1}), diag(t, t, t^{−1}, t^{−1}) .

The first two are the coroots of (e_{2} − e_{1}) and 2e_{1}, the last two those of 2e_{2} and e_{2} + e_{1}. Langlands and Shelstad ([L-S], (2.3)) define

x(σ_{T}) = ∏_{α∈R_{G}^{+},σ_{T}^{−1}α∈R_{G}^{−}}(a_{α})^{α}ˇ

for σ_{T} ∈ Γ using the a-data. This depends only on σ_{T} modulo Γ_{L}. It is one for the identity element and has therefore only to be computed for the involution of L/F. For the involution σ we get the product over all the four (co)roots

x(σ) = diag(t, t^{3}, t^{−1}, t^{−3})_{t=\sqrt{D_{0}}} .

2) The cocycle σ_{T} ↦ m(σ_{T}) = x(σ_{T})n(ω_{T} (σ_{T})) defined in [L-S], (2.3), has values in the \overline{F} valued points of the normalizer of T_{G}^{s}. Modulo the torus it coincides with σ_{T} ↦ h^{−σ}h. Therefore

σ_{T} ↦ m(σ_{T})h^{−σ}h ∈ T_{G}^{s}(\overline{F})

defines a 1-cocycle with values in T_{G}^{s}(\overline{F}). The action on this torus is understood to be the twisted one, which is transfered from the action of the Galois group on T_{G}(\overline{F}). It is determined by its value on σ, which is according to the computations above

λ = diag(t, t^{3}, −t^{−1}, −t^{−3})_{t=\sqrt{D_{0}}} .

The twisted action of σ on T_{G}^{s}(\overline{F}) is given by ξ^{σ} = w_{max}(σ(ξ)). Observe λ^{σ}λ = 1 (the cocycle condition with twisted action). The class of λ in H^{1}(Γ, T_{G}^{s}(\overline{F})) is λ_{T}.

The class λ_{T} of the cocycle defined by λ is trivial, if and only if there is a ρ ∈ T_{G}^{s}(\overline{F}) or

T_{G}^{s}(L), such that λ = w_{max}(σ(ρ))ρ^{−1}.

\underline{Remark 1}: With respect to the twisted action on T_{G}^{s}(\overline{F}) the map Int(h) defines a Galois equivariant isomorphism between T_{G}(\overline{F}) and T_{G}^{s}(\overline{F}). This gives λ_{T} as a cocycle with

values T_{G}(\overline{F}) alternatively. But H^{1}(Γ, T_{G}(\overline{F})) = H^{1}(Gal(L/F), T_{G}(\overline{F})) ≅ F^{∗}/Norm_{L/F} (L^{∗}) ≅ ZZ/2ZZ.

\underline{Remark 2}: The class λ_{T_{sc}} is defined as above with h_{der} instead of h. Therefore the image of λ_{T_{sc}} ∈ H^{1}(Γ, T_{sc}(\overline{F})) in H^{1}(Γ, T (\overline{F})) is λ_{T}. Both cocycles differ by a coboundary

z^{σ}z^{−1} ∈ B^{1}(Γ, Z(G)(\overline{F}) ⊂ B^{1}(Γ, T_{G}(\overline{F})).

4) Consider the root α_{1} : T_{G}^{s}(L) → L^{∗} defined by diag(ρ_{1}, ρ_{2}, κρ_{1}^{−1}, κρ_{2}^{−1}) ↦ ρ_{2}/ρ_{1}. Then

α_{1}(ρ^{σ}ρ^{−1}) = Norm_{L/F} (α_{1}(ρ))^{−1} ∈ F^{∗}. The cocycle condition for λ = diag(λ_{1}, λ_{2}, κλ_{1}^{−1}, κλ_{2}^{−1})

means κ = λ_{1}/σ(λ_{1}) = λ_{2}/σ(λ_{2}). Therefore α_{1}(λ) = λ_{2}/λ_{1} ∈ F^{∗}. Conversely, any λ_{2} ∈ F^{∗} can be prolongated to a cocycle λ by λ_{1} = 1 and κ = 1. Hence the cohomology class of λ is determined by α_{1}(λ) ∈ F^{∗}/Norm_{L/F} (L^{∗}). The latter is mapped isomorphically to ZZ/2ZZ under the character χ_{L/F}.

Therefore the cocycle λ, obtained above, has class λ_{T}

λ_{T} ∈ H^{1}(Γ, T_{G}(\overline{F}) ≅ ZZ/2ZZ

given by

λ_{T} = χ_{L/F} (α(λ)) = χ_{L/F} (D_{0}) .

In [L-S], (3.2) the factor Δ_{I}(η; (t_{1}, t_{2})) is defined to be ⟨λ_{T_{sc}}, s_{T}⟩. But s_{T} ∈ π_{0}((T̂_{G,sc})^{Γ}) is defined as the image of s ∈ T̂_{G}. In our case s ∈ (T̂_{M})^{Γ} holds. Therefore s_{T} is the image of the class s ∈ π_{0}(T̂_{G}^{Γ}) under the natural map induced from

i : T_{G,sc} → T_{G} .

Using functoriality of the pairing

⟨., .⟩ : H^{1}(Γ, T (\overline{F})) × π_{0}(T̂^{Γ}) → C^{∗}

with respect to i : T_{G,sc} → T_{G} we get ⟨λ_{T_{G,sc}}, s_{T} ⟩ = ⟨i_{∗}(λ_{T_{G,sc}}), s⟩. But i_{∗}(λ_{T_{G,sc}}) = λ_{T_{G}}.

Therefore we get from the calculations above

Δ_{I}(η; (t_{1}, t_{2})) = χ_{L/F} (D_{0}) .

The coboundary r_{p} for Ĝ

In [L-S], (2.5) one has to choose a p-gauge. Let Γ denote the absolute Galois group of F and let {1, ϵ} be the group generated by the reflection ϵ at the origin, both acting on the root system R_{G} of T_{G}. The product Σ of these two groups acts on R_{G} with four disjoint orbits, each consisting of a pair α, −α, where α runs over the positive roots R_{G}^{+}. So we can choose as a p-gauge the positive root gauge on each orbit. See [L-S],(2.1) and (2.5). Furthermore n = 1 and σ_{1} = id and w_{1} = id and u_{1}(w) = w in the notations of [L-S], (2.5).

In order to define the remaining Δ_{III_{2}}-factor one needs terms r_{p}(w). Before we compute them, we give a short review of some notations of [L-S], (2.5): Consider Weil groups. See also [T]. The Weil group W_{F} of F comes with a map π : W_{F} → Γ = Γ_{F} such that W_{L} = π^{−1}(Γ_{L}), where Γ_{L} denotes the absolute Galois group of L. Then W_{F} /W_{L} = Γ_{F} /Γ_{L} = Γ_{L/F}. The relative Weil groups are defined by W_{L/F} = W_{F} /[W_{L}, W_{L}]. One has exact sequences

0 → L^{∗} → W_{L/L} → Γ_{L/L} = 1
0 → L^{∗} → W_{L/F} → Γ_{L/F} → 0 .

We choose representatives v_{0} = 1 and v_{1} in W_{L/F} of 1 and σ in Γ_{L/F}. Then for u ∈ W_{±} = W_{L/F} we have

v_{0}(u) = \{ \begin{matrix}u & u ∈ W_{+} = W_{L/L} = L^{∗} \\ uv_{1}^{−1} & u ∉ W_{+} = L^{∗} . \end{matrix}

Now consider X = X_{∗}(T̂_{G}^{s}) = X^{∗}(T_{G}^{s}). The roots R = R_{X} are by definition the roots of T_{G}^{s} in G or the coroots of T̂_{G}^{s} in Ĝ. Consider X ⊗ C^{∗}. Then Langlands and Shelstad define

r_{p}(w)_{orbit} = ∏_{i=1}^{n} χ_{λ_{1}}(v_{0}(w))^{λ} = χ_{0}(v_{0}(w))^{λ}

for each orbit, i.e. positive coroot λ of T̂_{G}^{s}. Here we chose the ”positive” gauge on each orbit. The product over the contributions of all (four) orbits gives

r_{p}(w) = ∏_{λ∈R^{+}} χ_{0}(v_{0}(w))^{λ} ∈ X_{∗}(T̂_{G}^{s}, Ĝ) ⊗ C^{∗} = T̂_{G}^{s}(C)

for w ∈ W_{L/F}. Hence in our case

r_{p}(w) = diag(t, t^{3}, t^{−1}, t^{−3})_{t=χ_{0}(v_{0}(w))} ∈ T̂_{G}^{s} .

\underline{The definition of ξ_{χ}^{G}}:

Following Langlands and Shelstad [L-S], (2.6), the choice of r_{p}(ω) defines an L-extension

ξ_{χ}^{G} : ^{L}T → ^{L}G

of the given embedding

T̂_{G} → T̂_{G}^{s} → T_{G} ⊂ Ĝ ,

depending on the given χ-data. The L-extension of the given embedding is determined by the values

ξ_{χ}^{G}(1 × w) = r_{p}(w)nĜ(ω_{T} (w)) × w .

Here the L-groups are defined via the Weil groups as semi direct products Ĝ × W_{F} or

Ĝ × W_{L/F}.

The coboundary r_{p} for M̂

This section again refers to [L-S], (2.5). We repeat for M, what we have done for G.

Remember the definition of T̂_{M}^{s}, B̂_{M}^{s} for M̂. The inclusion ξ : M̂ → Ĝ maps the (positive) coroots of T̂_{M}^{s}, M̂ to the (positive) coroots of T̂_{G}^{s}, Ĝ. Our choice of χ-data for G therefore induces χ-data for M. In this sense

χ_{λ} = \{ \begin{matrix}χ_{0} & for λ a positive coroot of T̂_{M}^{s} \\ χ_{0}^{−1} & for λ a negative coroot of T̂_{M}^{s} \end{matrix} .

The positive coroots are given by β_{1}ˇ (t) = diag(t, 1, t^{−1}, 1) and by β_{2}ˇ (t) = diag(1, t, 1, t^{−1}).

For the corresponding positive roots β_{1} and β_{2} of T̂_{M}^{s} in M̂ we have

n(σ_{β_{1}}) = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

and

n(σ_{β_{2}}) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) .

The product of these matrices defines

nM̂(w_{max}) = ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} ) .

Consider again the group Σ generated by the reflection at the origin and Γ acting via the T_{M}-twisted action on the roots of T_{M}^{s} resp. the coroots of T̂_{M}^{s}. We have the two orbits β_{1}, −β_{1} and β_{2}, −β_{2}. We can therefore choose the positive R_{M}^{+} gauge on each orbit. Then n = 1 and σ_{1} = id and u_{1}(w) = w on each orbit in the notations of [L-S], (2.5). Consider X = X_{∗}(T̂_{M}^{s}, B̂_{M}^{s}, M̂) and R the set of coroots of T̂_{M}^{s}, B̂_{M}^{s}. The same formula as for G gives the corresponding factors for M. Hence these factors for M are

r_{p}(w) = diag(t, 1, t^{−1}, 1)diag(1, t, 1, t^{−1})_{t=χ_{0}(v_{0}(w))} = diag(t, t, t^{−1}, t^{−1})_{t=χ_{0}(v_{0}(w))} .

\underline{Definition of ξ_{χ}^{M}}:

Our choice of χ-data and gauge for M defines an extension

ξ_{χ}^{M} : ^{L}T_{M} → ^{L}M

of the given embedding

T̂_{M} → T̂_{M}^{s} → T_{M} ⊂ M̂ .

It factors over W_{L/F} and is defined by

ξ_{χ}^{M}(1 × w) = diag(t, t, t^{−1}, t^{−1})_{t=χ_{0}(v_{0}(w))}nM̂(ω_{T} (σ_{w})) × w .

The corestriction map

We will give a short review of the Langlands reciprocity law. Let T be a torus over F, split by the extension field L. We have the following chain of identifications

Hom_{cont}(T, C^{∗}) ≅ Hom_{cont}((L^{∗} ⊗ X_{∗}(T))^{Γ_{L/F}} , C^{∗}) ≅ Hom(L^{∗}, X^{∗}(T) ⊗ C^{∗})_{Γ_{L/F}}
≅ Hom(L^{∗}, X_{∗}(T̂) ⊗ C^{∗})_{Γ_{L/F}} ≅ Hom(L^{∗}, T̂(C))_{Γ_{L/F}} .

We have a commutative diagram

\begin{aligned}Hom_{cont}(T, C^{∗}) & ≅ & Hom(L^{∗}, T̂(C))_{Γ_{L/F}} \\ ↑ & & ↑ \\ H_{cont}^{1}(W_{L/F} , T̂(C)) & ← & Hom(L^{∗}, T̂(C)) \end{aligned} .

The left vertical isomorphism is the ”Langlands” reciprocity map for tori. It is functorial in T. The right vertical map is the obvious surjection. The lower horizontal map is the corestriction map. For the details see e.g. Milne [M], p.135ff and thm.8.6. The corestriction map is defined as follows. For f ∈ Hom(L^{∗}, T̂(C)) one has cor(f) ∈ H^{1}(W_{L/F} , T̂(C)), defined by the cocycle

cor(f)(w) = ∑_{g} w_{g}f(w_{g}^{−1}ww_{g′}) ,

where w_{g} are representatives of Γ_{L/F} in W_{L/F} and ww_{g′} = w_{g} mod L^{∗} determines g′ resp. w_{g′} for given w and g. For the details see [M], p.129.

Let us specialize to the quadratic extension field L over F. We have already specified representatives w_{1} = v_{0} = 1 and w_{σ} = v_{1} in W_{L/F}. Let denote θ = v_{1}^{2}. Observe θ ∈ F^{∗}. We get

cor(f)(λ) = f(λ)v_{1}f(v_{1}^{−1}λ)v_{1}) = f(λ)v_{1}f(λ^{σ}) , λ ∈ L^{∗} ⊂ W_{L/F}

cor(f)(λv_{1}) = f(λv_{1} · v_{1})v_{1}f(v_{1}^{−1}λv_{1}) = f(λθ)v_{1}f(λ^{σ}) , λ ∈ L^{∗} ⊂ W_{L/F} .

The torus T = N_{L}^{1}

In this section we make the Langlands reciprocity map explicit in the simplest example T = N_{L}^{1} of the norm-1 group of a quadratic extension field.

Notations: Let N_{L}^{1} be the one dimensional torus of elements of norm one in L^{∗}. The dual group of T = N_{L}^{1} will be denoted T̂ = C_{−}^{∗}. This notation indicates, that as abstract group T̂ is C^{∗}, but the Galois or Weil group acts via z^{σ} = z^{−1}, z ∈ C_{−}^{∗}. Let σ denote the generator of Gal(L/F).

Every cocycle a ∈ Z^{1}(W_{F} , N̂_{L}^{1}) is a character, when restricted to W_{L}. It is therefore trivial on [W_{L}, W_{L}] and factors over W_{L/F}

0 → L^{∗} → W_{L/F} → Γ_{L/F} → 0 .

Choose v_{1} ∈ W_{L/F}, but v_{1} ∉ L^{∗}. Then v_{1}(v_{1}^{2})v_{1}^{−1} = (v_{1})^{2} ∈ F^{∗}. One can show v_{1}^{2} ∉ Norm_{L/F} (L^{∗}). An easy computations shows, that the cocycle conditions for a are equivalent to:

1) The restriction of a to L^{∗} is a continuous character with values in C_{−}^{∗}.

2) a(σ(λ)) = a(λ^{−1}) for λ ∈ L^{∗}.

3) a(v_{1}^{2}) = a(v_{1})v_{1}(a(v_{1})) = a(v_{1})/a(v_{1}) = 1. Therefore condition 2) and 3) are equivalent to a(λ) = 1, λ ∈ F^{∗}. If we modify a by a coboundary, this does not change the restriction of a to L^{∗}, whereas the restriction to L^{∗} ·v_{1} can in this way be multiplied by an arbitrary constant in C^{∗}. Therefore we formulate the last cocycle condition as

4) a(λv_{1}) = const · a(λ). The cohomology class [a] of a is therefore uniquely determined by the restriction of a to L^{∗}, which is an element of in Hom_{cont}(L^{∗}/F^{∗}, C^{∗}).

Now let us make the Landlands reciprocity map explicit. Suppose, we are given a continuous character χ ∈ Hom_{cont}(N_{L}^{1}(F), C^{∗}). We can extend χ to a continuous character χ′ : L → C^{∗} via the embedding N_{L}^{1}(F) ⊂ L^{∗}. The map

Hom(L^{∗}, N̂_{L}^{1}) → Hom(L^{∗}, N̂_{L}^{1})_{Γ} ≅ Hom((L^{∗} ⊗ ZZ)^{Γ}, C^{∗}) ≅ Hom(N_{L}^{1}(F), C^{∗})

is readily seen to be given by the restriction of the character

(χ′ : L^{∗} → C^{∗}) ↦ (χ′|N_{L}^{1}(F) : N_{L}^{1}(F) → C^{∗}) .

This describes the upper way through the diagramm

\begin{aligned}Hom_{cont}(N_{L}^{1}(F), C^{∗}) & ≅ & Hom(L^{∗}, C_{−}^{∗}))_{Γ_{L/F}} \\ ↑ & & ↑ \\ H_{cont}^{1}(W_{L/F} , C_{−}^{∗})) & ← & Hom(L^{∗}, C_{−}^{∗}) \end{aligned} .

It maps χ′ to χ. The lower horizontal map, the corestriction map, was already explained above. It maps χ′ to a cohomology class cor(χ′), which is completely determined by its ’restriction’ to L^{∗} = W_{L}^{ab}. This restriction is the character

cor(χ′)(λ) = χ′(λ)v_{1}(χ′(λ^{σ}) = χ′(λ/λ^{σ}) , λ ∈ L^{∗} = W_{L}^{ab} .

Note that x = λ/λ^{σ} ∈ N_{L}^{1}(F). Conversely every element x ∈ N_{L}^{1}(F) can be written in this form, λ ∈ L^{∗} being unique up to a multiple from F^{∗}. Therefore the Langlands reciprocity in this case is explicitely given by

Hom_{cont}(N_{L}^{1}(F), C^{∗}) ← χ ↦ a(λ) = χ(λ/λ^{σ}) ∈ H^{1}(W_{F} , C_{−}^{∗}) .

Notation explains itself by the fact, that the cocycle class [a] is determined by its restric-

tion a(λ) ∈ Hom_{cont}(W_{L}^{ab}, C_{−}^{∗}) for λ ∈ W_{L}^{ab} ≅ L^{∗}.

\underline{Example 1}: Let s ∈ ˆ((N_{L})^{1})^{Γ} = \{±1\}. Consider the homomorphism ord : W_{F} → W_{F}^{ab} ≅ F^{∗} →^{ord} ZZ. It defines a 1-cocycle t^{s}(w) = s^{ord(w)} in Z^{1}(W_{F} , N̂_{L}^{1}). The map

L^{∗} ≅ W_{L}^{ab} → W_{F}^{ab} ≅ F^{∗} ,

induced by the inclusion map W_{E} ↪ W_{F} is the norm map Norm_{L/F} : L^{∗} → F^{∗}. See [Ta], (1.2.2). The restriction to W_{L}^{ab} of the cocycle t^{s} is therefore t^{s}(λ) = (−1)^{ord(Norm_{L/F} (λ))} Under Langlands reciprocity map t^{s} corresponds to a continuous character χ(x) of N_{L}^{1}(F) given by

L/F unramified: χ is trivial.

L/F is ramified: χ(x) = (−1)^{ord(λ)} where x = λ/λ^{σ}. If the residue characteristic is

odd, then χ(x) = x mod π_{L}o_{L} ∈ {±1}.

\underline{Example 2}: Put χ′ = χ_{0}, where χ_{0} is the character of L^{∗}, used in the definition of χ-data. Consider its restriction to N_{L}^{1}(F) ⊂ L^{∗}, also denoted χ = χ_{0}. Since χ_{0}(x^{σ}) = χ_{0}(x)^{−1} we get

cor(χ)(λ) = χ_{0}^{2}(λ) , λ ∈ L^{∗} ≅ W_{L}^{ab} .

Therefore the character χ_{0} ∈ Hom_{cont}(N_{L}^{1}(F), C^{∗}) corresponds via Langlands reciprocity to the class of the cocycle a(w) = χ_{0}^{2}(v_{0}(w)) of W_{L/F} in N̂_{L}^{1}.

Comparison of L-homomorphisms

In this section we compute Δ_{III_{2}} defined in [L-S], (3.5).

Observe, that T_{M} is identified with T_{G} as in lemma 1. This corresponds to an identification of T̂_{M} and T̂_{G} and also ^{L}T_{M} and ^{L}T_{G}.

Using this identification we get two different homomorphisms from ^{L}T_{M} = ^{L}T_{G} to ^{L}G, namely ξ ∘ ξM̂ ; χ and ξĜ ; χ. They agree on T̂_{M} = T̂_{G}, hence differ by a 1-cocycle a(w) of W_{L/F} with values in T̂_{G} (twisted Galois action induced from T̂_{G}!)

ξ ∘ ξM̂ ; χ(w) = a(w) · ξĜ ; χ(w) , w ∈ W_{L/F} .

\underline{Lemma 3}: Let χ_{0} be the character specified in our choice of χ-data. Then

a(w) = (−1)^{i(w)}diag(1, χ_{0}^{−2}(v_{0}(w)), 1, χ_{0}^{2}(v_{0}(w))) ∈ T̂_{G} .

By definition it satisfies a(w_{1}w_{2}) = a(w_{1})a(w_{2})^{nĜ(σ(w_{1}))}.

\underline{Proof}: Using our formulas for ξ_{χ}^{M} and ξ_{χ}^{G} we compute a(w) as follows: Let i(w) be 0 or 1 according whether w ∈ L^{∗} ⊂ W_{L/F} or not. Then

diag(t, t, t^{−1}, t^{−1})_{t=χ_{0}(v_{0}(w))} ( \begin{matrix}0 & E \\ −E & 0 \end{matrix} )^{i(w)}

= a(w) · diag(t, t^{3}, t^{−1}, t^{−3})_{t=χ_{0}(v_{0}(w))} ( \begin{matrix}0 & −E \\ E & 0 \end{matrix} )^{i(w)} .

The matrix cancels, and we obtain the claim.

We are interested in the corresponding cohomology class [a] of a(w) in H^{1}(W_{L/F} , T̂_{G}). [a] can be transfered to a cohomology class

a_{T} ∈ H^{1}(W_{L/F} , T̂_{G}) .

By Langlands reciprocity a_{T} defines a character ⟨a_{T} , .⟩ in Hom_{cont}(T_{G}(F), C^{∗}). By definition ([L-S],(3.5))

Δ_{III_{2}}(η, t_{1}, t_{2}) = ⟨a_{T} , η⟩

2) Consider the isomorphisms T̂_{G}^{s} → T̂_{G} → T̂(L^{∗}) × T̂(N_{L}^{1}) given by the composite of the maps T̂_{G}^{s} → T_{G}

diag(w_{1}, w_{2}, w′_{1}, w′_{2}) mod (t, t, t^{−1}, t^{−1}) ↦ diag(z_{1}, z_{4}, z_{2}, z_{3}) = diag(w_{2}w′_{1}, w_{1}w′_{1}, w_{1}w′_{2}, w_{2}w′_{2}) and T̂_{G} → T̂(L^{∗}) × T̂(N_{L}^{1})

diag(z_{1}, z_{4}, z_{2}, z_{3}) ↦ (z_{1}, z_{2}) × z_{3}/z_{2} = (w_{2}w′_{1}, w_{1}w′_{2}) × w_{2}/w_{1} .

The latter has inverse ˆ(L^{∗}) × T̂(N_{L}^{1}) → T̂_{G}

(u, v) × w ↦ (u, uw^{−1}, v, vw) ∈ T̂_{G} .

Here T (L^{∗}) and T (N_{L}^{1}) are the F-tori attached to the multiplicative group L^{∗} respectively the group N_{L}^{1} of elements of norm one. Both maps are isomorphisms, the second because of z_{1}z_{2} = z_{3}z_{4}. Endow T_{G}^{s} with the twisted Galois action and T̂_{G} with the induced twisted action. It agrees with the twisted action transfered from T̂_{M} to T̂_{M} = T̂_{G}. With these conventions, the isomorphisms above are equivariant with respect to the Weil groups.

3) The dual isomorphism L^{∗} × N_{L}^{1} → T_{G}^{s} is given by

(y, z) ∈ L^{∗} × N_{L}^{1} ↦ diag(x_{1}, x′_{1}, x_{2}, x′_{2}) = diag(y^{σ}z^{−1}, y, yz, y^{σ}) ∈ T_{G}^{s}(\overline{F}) .

The inverse T_{G}^{s} → N_{L}^{1} therefore maps diag(x, x′, x^{σ}, (x′)^{σ}) to x^{σ}/x′. The composite π : T_{G} → T_{G}^{s} → N_{L}^{1} is defined over F. The first map is Int(h^{−1}). It maps η ∈ T_{G}(F) to

diag(x, x′, x^{σ}, (x′)^{σ}) ∈ T_{G}^{s}(\overline{F}). We obtain

π : T_{G} → N_{L}^{1} , π(η) = x^{σ}/x′ .

4) Resume: The cohomology class [a] ∈ H^{1}(W_{L/F} , T̂_{G}) is the product of two classes [a_{1}] and [a_{2}]. One is attached to the cocycle

a_{1}(w) = diag(1, χ_{0}(v_{0}(w))^{−2}, 1, χ_{0}(v_{0}(w))^{2}) ∈ T̂_{G} , w ∈ W_{L/F} ,

which is the image of

b(w) = χ_{0}(v_{0}(w))^{2} ∈ Z^{1}(W_{L/F} , T̂(N_{L}^{1}))

under the composition of the inclusion T̂(N_{L}^{1}) into the second factor of T̂(L^{∗}) × T̂(N_{L}^{1}) and the isomorphism with T̂_{G}. The second is the class of the cocycle a_{2}(w) = (−1)^{i(w)} · id of T̂_{G}. It is the image of the class of the cocycle ((−1)^{i(w)}, (−1)^{i(w)}) of T̂(L^{∗}) under the inclusion into the first factor of T̂(L^{∗}) × T̂(N_{L}^{1}) and the isomorphism with T̂_{G}. This is a

coboundary in T̂(L^{∗}), namely ((−1)^{i(w)}, (−1)^{i(w)}) = (1, −1)^{w}(1, −1)^{−1}. Hence [a] = [a_{1}].

5) Rember the map π : T_{G} → T (L^{∗}) × N_{L}^{1} → N_{L}^{1}. In 4) we have shown a_{T} = π̂_{∗}([b]). By functoriality of Langlands reciprocity we therefore get ⟨a_{T} , η⟩ = ⟨π̂_{∗}([b]), η⟩ = ⟨b, π(η)⟩. By lemma 3 ⟨b, .⟩ is the character χ_{0} of N_{L}^{1}(F). Therefore

Δ_{III_{2}}(η; (t_{1}, t_{2})) = χ_{0}(x^{σ}/x′) .

Appendix, The archimedian case

From lemma 2 and the remark thereafter we got

Δ(η, t_{1}, t_{2}) = (χ_{L/F} |.|)((x − x^{σ})(x′ − (x′)^{σ}))|xx′|^{−1}

with the assignments x = t_{1}t_{2} and x′ = t_{1}t_{2}^{σ}, where η is defined as

η = ( \begin{matrix}a & 0 & bD_{0} & 0 \\ 0 & a′ & 0 & b′D_{0} \\ b & 0 & a & 0 \\ 0 & b′ & 0 & a′ \end{matrix} ) ,

where x = a + b\sqrt{D_{0}} and x′ = a′ + b′\sqrt{D_{0}}. From [L-S] these local transfer factor satisfy the global hypothesis: Their product over all local places is one, for global data D_{0}, t_{1}, t_{2}.

If we consider the problem of stabilisation of the trace formula as in [Ko2], we assume that the adelic transfer factor satisfies the global hypothesis (loc cit p.178). For this we normalize the archimedian transfer factor such that it is the one given in [L-S]. The only case of elliptic tori in the archimedian case are the ones of case 1, studied above. The nonelliptic case is rather easy, and will be skipped.

Warning: Using this normalisation we therefore do not a priori follow the conventions of [Ko2] p.184, when we define the terms Δ_{∞}(γ_{M}, γ), nor do we then follow [Ko2] p.186, when we define h_{∞}^{M}. Having fixed our choice of archimedian transfer factor, we are forced to define h_{∞}^{M} such, that it satisfies [Ko2] formula (7.4)

SO_{(t_{1},t_{2})}(h_{∞}^{M}) = ⟨β(η), s_{M}⟩Δ_{∞}(η, (t_{1}, t_{2}))e(I)trξC(η) · vol^{−1} .

Our choice of archimedian transfer factor may result in a change of h_{∞} defined on [Ko2], p.186 by a constant. It is important to determine this constant. For this it is enough to consider the (only nontrivial) case 1 with D_{0} = −1 and regular semisimple elliptic (t_{1}, t_{2}) ∈ C^{∗} with |t_{1}| = |t_{2}| = 1. Then χ_{L/R}|.| = id, and we have by assumption

Δ_{∞}(η, t_{1}, t_{2}) = (x − \overline{x})(x′ − \overline{(}x′))(x\overline{x})^{−1} .

With the definition of x and x′ in terms of t_{i} this is (t_{1}t_{2} − (t_{1}t_{2})^{−1})((t_{1}/t_{2}) − (t_{2}/t_{1})) or equivalently (t_{1}^{2} + 1 + t_{1}^{−2}) − (t_{2}^{2} + 1 + t_{2}^{−2}). Thus

Δ_{∞}(η, t_{1}, t_{2}) = (tr(τ_{2} ⊗ τ_{0}) − tr(τ_{0} ⊗ τ_{2}))(γ) , γ = (t_{1}, t_{2}) .

τ_{k} is the representation of SU(2, R) on Symm^{k}(C^{2}). Suppose that the representation ξ_{C} is fixed. If the holomorphic discrete series attached has weights k_{1} ≥ k_{2} ≥ 3. Then put r_{1} = k_{1}+k_{2}−2 and r_{2} = k_{1}−k_{2}+2. Recall, that the Langlands parameter μ_{1} = (k_{1}+k_{2}−3)/2 and μ_{2} = (k_{1} −k_{2} +1)/2 correspond to a pair of discrete series representations of Gl(2, R) of weights r_{1} = k_{1} +k_{2} −2, r_{2} = k_{1} −k_{2} +2. The Langlands parameter define a homomorphism of C^{∗} to T̂_{M} whose composite with the admissible homomorphism as in Lemma 1 defines a homomorphism of C^{∗} to Ĝ with cocharacters of weights μ_{1}+μ_{2} = k_{1}−1 and μ_{1}−μ_{2} = k_{2}−2. This belongs to the L-packet of the holomorphic discrete series of weight (k_{1}, k_{2}).

\underline{Lemma}: Consider the z-extension M_{1} = Gl(2, R)^{2}. Let h(π_{r_{1}} ⊗ π_{r_{2}} denote pseudo coefficients for (holomorphic) discrete series of weights r_{1} and r_{2} on M_{1}. Then with our choice of Δ_{∞}(η, (t_{1}, t_{2}) and under the assumptions above the function

h_{∞}^{M_{1}} = −(h(π_{r_{1}} ⊗ π_{r_{2}}) − h(π_{r_{2}} ⊗ π_{r_{1}})) , (r_{1} > r_{2})

satisfies formula (7.4) of [Ko2].

\underline{Proof}: From [Ko2], p.186 and the fact, that our h_{∞}^{M_{1}} is proportional to the one defined in loc. cit this is true up to some universal constant (independent from ξ_{C}). We can therefore assume, that ξ_{C} is the trivial representation. This corresponds to the case

(k_{1}, k_{2}) = (3, 3) and (r_{1}, r_{2}) = (4, 2). By our assumptions trξC(η) = 1 and e(I) = 1 holds in

formula (7.4). Because we replaced M by M_{1}, the notions of stable orbital integrals and ordinary orbital integrals coincide. Furthermore according to [CAP], §11

SO_{(t_{1},t_{2})}^{M_{1}}(h(π_{4} ⊗ π_{2}) − h(π_{2} ⊗ π_{4}))(t_{1}, t_{2})
= vol^{−1} · [(−tr(τ_{2}(t_{1}))(−tr(τ_{0}(t_{2})) − (−tr(τ_{0}(t_{1}))(−tr(τ_{2}(t_{2}))]
= vol^{−1} · Δ_{∞}(η, (t_{1}, t_{2})) .

Note, that tr(τ(t)) = tr(τ^{∗}(t)) if |t| = 1. Thus is suffices to show

⟨β(η), s_{M}⟩ = −1

in order to complete the proof of the lemma.

Recall the definition of β(η) ∈ X^{∗}(Ĝ_{η})^{Γ(∞)} from [Ko2], p.182 and p.167. It depends on η only via the elliptic torus I = G_{η} ⊂ G. Namely conjugate the structure homomorphism defining the Shimura variety, such that it maps into I. For our choices of η and our choice of structure homomorphism, this amounts to conjugation by the identity. The the (conjugated) structure homomorphism defines a homomorphism μ_{h} : G_{m}/C → I/C as in [Ko2], p.167. In our case (after a further conjugation in G(C)) we get a conjugate

μ : G_{m}/C → T_{G}^{s}/C ↪ G/C

μ(z) = diag(z, z, 1, 1) .

This defines a cocharacter μ ∈ X_{∗}(T_{G}^{s}). In the notations used above this cocharacter μ corresponds to the element

(m_{1}, m_{2}, m′_{1}, m′_{2}) = (1, 0, 1, 0) ∈ X_{∗}(T_{G}^{s}) .

On the other hand, the element s = s_{M} = (1, −1, 1, −1) ∈ T̂_{G} from the endoscopic data corresponds to (w_{1}, w_{2}, w′_{1}, w′_{2}) = (−1, 1, 1, −1) ∈ T̂_{G}^{s} = X^{∗}(T_{G}^{s}) ⊗ C^{∗}. (use the isomorphism T̂_{G} ≅ T̂_{G}^{s} specified above). The pairing ⟨μ, (w_{1}, w_{2}, w′_{1}, w′_{2})⟩ = w_{1}w′_{1} therefore gives ⟨μ, s⟩ = −1. This proves the lemma.

References

[M] Milne J.S., Arithmetic duality theorems, perspectives of mathematics, vol. 1, academic press (1986)

[L-S] Langlands R. - Shelstad D., On the definition of transfer factors, Math. Ann. 278 (1987), 219 - 271

[W] Weissauer R., A special case of the fundamental lemma I, (the case GSp(4, F)), preprint Mannheim 1993

A special case of the fundamental lemma IV

The twisted case

Rainer Weissauer

University of Mannheim, July 94

1.Unramified characters:

We consider the embedding of the endoscopic group ^{L}M into ^{L}G as in [W1]. Also we maintain the choices of tori and Borel groups B_{M} ⊂ M and B ⊂ G as they were made in [W1]. B_{M} is the Borel group, consisting of all matrices

( \begin{matrix}t_{1} & ∗ \\ 0 & t′_{1} \end{matrix} ) × ( \begin{matrix}t_{2} & 0 \\ ∗ & t′_{2} \end{matrix} ) .

modulo (t · id, t^{−1} · id) for t ∈ F^{∗}.

Let T_{M}^{s} be the split torus in M, which consists of the diagonal matrices in B_{M} of the

form t = (diag(t_{1}, t′_{1}), diag(t_{2}, t′_{2})) modulo the matrices of the form (diag(t, t), diag(t^{−1}, t^{−1}))

for t ∈ F^{∗}. Then the positive roots are R_{M}^{+} given by the two characters χ_{1}(t) = t_{1}/t′_{1} and χ_{2}(t) = t′_{2}/t_{2}. Let us consider the split torus of diagonal matrices T_{G}^{s} ⊂ GSp(4, F). Here we label the elements in the form

( \begin{matrix}x_{1} & 0 & ∗ & ∗ \\ ∗ & x_{2} & ∗ & ∗ \\ 0 & 0 & x′_{1} & ∗ \\ 0 & 0 & 0 & x′_{2} \end{matrix} ) ,

where the dots indicate our fixed choice of Borel group B in GSp(4, F).

With these notations the following maps define an isomorphism between the two split tori T_{M}^{s} and T_{G}^{s}

(diag(t_{1}, t′_{1}), diag(t_{2}, t′_{2}))modF^{∗} ↦ diag(t′_{2}t′_{1}, t′_{2}t_{1}, t_{2}t_{1}, t_{2}t′_{1})

and conversely

diag(x_{1}, x_{2}, x′_{1}, x′_{2}) ↦ diag(x_{2}, x_{1}) × diag(x′_{1}/x_{2}, 1) mod F^{∗} .

This is an admissable isomorphism. We may simply copy the argument of the first section of [W3].

Under this admissable isomorphism, unramified characters of the two split tori correspond to each other. The character

(ξ, η, ξ′, η′)(diag(t_{1}, t′_{1}) × diag(t_{2}, t′_{2}))= ξ(t_{1})η(t′_{1})ξ′(t_{2})η′(t′_{2})

with ξη = ξ′η′ corresponds to

diag(x_{1}, x_{2}, x′_{1}, x′_{2}) ↦ ξ(x_{2})η(x_{1})ξ′(x′_{1}/x_{2})η′(1) .

If we use the notation

(χ_{1} × χ_{2} × χ)(diag(x_{1}, x_{2}, x′_{1}, x′_{2})) = χ_{2}(x_{1})χ_{1}(x_{2})χ(x_{1}x′_{1})

we get the correspondence

η = χχ_{2} , ξ/ξ′ = χ_{1} , ξ′ = χ ,

and

χ = ξ′ , χχ_{1} = ξ , χχ_{2} = η , χχ_{1}χ_{2} = η′ .

This relation defines a map from the set of isomorphism classes of unramified principal series representations of M

Ind( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )^{Gl(2,F)}(ξ, η) ⊗ Ind( \begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix} )^{Gl(2,F)}(ξ′, η′)

to the set of isomorphism classes of unramified principal series representations

χ_{1} × χ_{2} ◁ χ = Ind_{B(F)}^{G(F)}(χ_{1} × χ_{2} × χ)

where χ_{1} = ξ/ξ′ , χ_{2} = η/ξ′ , χ = ξ′. This corresponds to the following relation between Satake parameters. Put

α = ξ(π_{F}) , β = η(π_{F})
α′ = ξ′(π_{F}) , β′ = η′(π_{F}) with αβ = α′β′.

Furthermore put

α_{0} = χ(π_{F}), α_{1} = χ_{1}(π_{F}), α_{2} = χ_{2}(π_{F}).

Then α_{0} = α′, α_{0}α_{1} = α, α_{0}α_{2} = β, α_{0}α_{1}α_{2} = β′. The above correspondence between isomorphism classes of principal series representations is related to the L-embedding

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) × ( \begin{matrix}α′ & 0 \\ 0 & β′ \end{matrix} ) ∈ T̂_{M}^{s} ≅ T̂_{M} ↪ ^{L}M ↪ ( \begin{matrix}α & 0 & 0 & 0 \\ 0 & β′ & 0 & 0 \\ 0 & 0 & β & 0 \\ 0 & 0 & 0 & α′ \end{matrix} )

= ( \begin{matrix}α_{0}α_{1} & 0 & 0 & 0 \\ 0 & α_{0}α_{1}α_{2} & 0 & 0 \\ 0 & 0 & α_{0}α_{2} & 0 \\ 0 & 0 & 0 & α_{0} \end{matrix} ) ∈ T̂_{G} .

This element in T̂_{G} is uniquely defined only up to the action of the Weyl groups of the tori. We may make the following changes: We are allowed to interchange α_{1} and α_{2} and leave α_{0} unchanged. Or we may replace α_{0} by α_{0}α_{2}, α_{2} by α_{2}^{−1} and leave α_{1} unchanged. These two substitutions generate the Weyl group W_{G} of T̂_{G}.

2.Restriction of scalars

1) Consider a reductive group G′ over F. Then G = Res_{E/F} (G′) has L-group

LG = I_{W_{E}}^{W_{F}} (Ĝ′) ◁ W_{F}

where ^{L}G′ = Ĝ′ ◁ W_{F} in the sense of [Borel], p.35 (2). Remember, for a group X with W_{E} action

I_{W_{E}}^{W_{F}} (X) = {f : W_{F} → X | f(w′w) = w′ · f(w), w ∈ W_{F} , w′ ∈ W_{E}} .

W_{F} act by (r_{w}f)(x) = f(xw). If X = X^{∗}(T′) for a torus T′, then I_{W_{E}}^{W_{F}} (X) = X^{∗}(Res_{E/F} (T′))

(loc.cit).

X can be equivariantly embedded into I_{W_{E}}^{W_{F}} (X). To x one assigns the map w ↦ w · x.

2) Now assume that W_{E} to be a normal subgroup of W_{F} of index n, with cyclic factor group generated by σ_{F} ∈ W_{F}. Furthermore assume X is already a W_{F} -module, whose W_{F} module structure is given by restriction. Then put

(θf)(x) = σ_{F}^{−1}f(σ_{F} x) .

Then θ preserves I_{W_{E}}^{W_{F}} (X), commutes with the action of W_{F} and satisfies θ^{n} = id.

I_{W_{E}}^{W_{F}} (X)^{θ} = X, namely θ-invariance implies f(wx) = w · f(x) for w = σ_{F}. As σ_{F} and W_{E} generate W_{F}, therefore f(w) = w · f(1) for all w ∈ W_{F}.

Suppose X is abelian. Then f(x) ↦ ∑_{i=1}^{n}(θ^{i}f)(1) ∈ X is a W_{F} equivariant norm map, which annihilates all f(x) − (θf)(x). If we put f_{x}(w) = w · x for w ∈ W_{E} and zero else, then f_{x} ∈ I_{W_{E}}^{W_{F}} (X). This defines a splitting (respecting only the action of W_{E}). Therefore the norm map is surjective. It induces an isomorphism I_{W_{E}}^{W_{F}} (X)_{θ} ≅ X. To show this, it is enough to show that the kernel is generated by the f(x) − (θf)(x). This is an easy exercise and most easily seen using the isomorphism Ψ, described below.

3) One defines θ : ^{L}G → ^{L}G when G = Res_{E/F} (G′), using the description of ^{L}G as induced group given above. This is functorial for G′_{1} → G′_{2}. It is sometimes useful, to rewrite things by the isomorphism

Ψ : Ĝ ⟶ Ĝ′ × .... × Ĝ′

defined as

Ψ(f) = (f(1), (θf)(1), ..., (θ^{n−1}f)(1)) .

It is clear that

Ψ(θf) = ((θf)(1), (θ^{2}f)(1), ..., f(1)) = θ̂(Ψ(f)) ,

where θ̂(g_{1}, .., g_{n}) = (g_{2}, ..., g_{n}, g_{1}).

Ψ(r_{w}(f)) = w · Ψ(f) = (1 ◁ w)(Ψ(f) ◁ 1)(1 ◁ w^{−1}) , w ∈ W_{F} .

As σ_{F} and W_{E} generate W_{F}, it is enough to check this for w = σ_{F} and w ∈ W_{E}. In both cases this formula is clear by definition, namely

Ψ(r_{w}(f)) = (f(w), (θf)(w), ..., (θ^{n−1}f)(w)) = (w(f(1)), w((θf)(1)), ..., w((θ^{n−1}f)(1))) ,

for w ∈ W_{E} and

Ψ(r_{σ_{F}})(f)) = (f(σ_{F}), (θf)(σ_{F}), ..., (θ^{n−1}f)(σ_{F})) = (σ_{F} (θf(1)), σ_{F} ((θf)^{2}(1)), ..., σ_{F} (f(1))) ,

where

Ψ(f) = (f(1), (θf)(1), ..., (θ^{n−1})(1)) .

Therefore the contruction of θ, given above, coincides with the one used by Kottwitz in terms of θ̂ using the isomorphism Ψ.

4) As a consequence we get in the special case I_{W_{E}}^{W_{F}} (X^{∗}(T′)) = X^{∗}(Res_{E/F} (T′)), where E/F is an unramified field extension: X^{∗}(Res_{E/F} (T′)_{θ}) = X^{∗}(Res_{E/F} (T′))^{θ} ≅ X^{∗}(T′). Henceforth

Res_{E/F} (T′)^{θ} ≅ T′

and

Res_{E/F} (T′)_{θ} ≅ T′ .

If we compose this isomorphism with N_{T,θ} : Res_{E/F} (T′) → Res_{E/F} (T′)_{θ}, we get a natural map

Nm_{E/F} : Res_{E/F} (T′) → T′

and from this a map

Norm_{E/F} : T′(E) → T′(F) .

The map Nm_{E/F} is given, for x ∈ Res_{E/F} (T′)(\overline{F}) = T′(⨁_{1}^{n} \overline{F}), by pr_{1}(∏_{0}^{n−1} θ^{i}(x)) ∈ T′(\overline{F}).

\underline{List of Norms}: We distinguish between the following norm maps:

N_{T,θ} : Res_{E/F} (T′) → Res_{E/F} (T′)_{θ}

Nm_{E/F} : Res_{E/F} (T′) → Res_{E/F} (T′)_{θ} ≅ T′

Norm_{E/F} : T′(E) → T′(F)

norm : Res_{E/F} (T′) → T′ ≅ T_{M}, the first being the map Nm_{E/F}, the second the admissable isomorphism.

N_{θ}(δ) = δθ(δ)...θ^{n−1}(δ) = (δθ)^{n}.

N_{σ_{F}} (δ) = δσ_{T} (δ)...σ_{T}^{n−1}(δ) = (δσ_{F})^{n}σ_{F}^{−n}.

Note N_{σ_{F}} (δ) = N_{θ}(δ) holds for δ ∈ G(F) = G′(E).

3.Twisted endoscopy:

Let F be a local nonarchimedian field of characteristic zero and let E/F be an unramified cyclic field extension of degree n. Let G be the Weil restriction Res_{E/F} (GSp(4, E)). This is an algebraic group over F. The Frobenius automorphism, a generator of the Galois group of E/F, acts by an algebraic automorphism θ on G. This action is defined over F. We are in the situation of [KS], where one can study the twisted endoscopic groups attached to (G, θ). In [KS] several definitions are necessary. In our special case, most of these definitions turn out to be either rather easy or unneccessary.

First of all, that group G is quasisplit. The automorphism θ respects a pair (B, T). This implies G = G^{∗}, θ = θ^{∗} and ψ = id in the sense of [KS], (1.2). The fixed group T^{θ} = T_{d} is a maximal F-split torus in G.

The restricted root system R_{res}(G, T) can be identified with the root system R(G^{θ}, T^{θ}) of the split group GSp(4, F). Especially it is reduced, all its roots are of type R_{1} in the sense of [KS], (1.3).

The dual group Ĝ is ∏_{i=1}^{n} GSp(4, C), the Galois group acts via the quotient group Gal(E/F), the Frobenius acts like the dual automorphism

θ̂(x_{1}, ..., x_{n}) = (x_{2}, ..., x_{n}, x_{1}) .

Let us consider the endoscopic datum (M,^{L}M, s, ξ) defined as follows. Let M be the group Gl(2, F)^{2}/F^{∗}. This is the group considered in [W1] up to the change of notation, M instead of M. The new notation shall indicate, that we consider a different endoscopic datum. In the case E = F the embedding of ^{L}M into ^{L}G will not be the one considered in [W1]. However, in the case E = F, the new endoscopic datum will be equivalent to the one considered in [W1].

Remember that in [W1] we had fixed an embedding of M̂ into Ĝ^{θ} = GSp(4, C). In order to define the new embedding, identify M̂ and M̂. Via the diagonal embedding we can therefore embed M̂ into Ĝ = ∏ GSp(4, C) using the embedding of [W1]. Let σ be the Frobenius automorphism in Gal(E/F). Then the new embedding ξ : ^{L}M → ^{L}G is defined by

ξ(h × 1) = (h, h, .., h) ◁ 1
ξ(1 × σ) = s ◁ σ = (1, ..., 1, diag(1, −1, 1, −1)) ◁ σ .

The image ξ(M̂) is the θ̂ centralizer of the element s, defined in the last equation.

We want to proof the endoscopic fundamental lemma for this embedding. We follow the method of Labesse and consider elementary functions. This allows to reduce the proof to the case of the unit elements of the Hecke algebra. For the comparison of elementary functions on M and G one needs the transfer factors for the split torus T_{M}^{s} in M. For the other tori we need not compute these transfer factors explicitly. It is enough to relate the transfer factors in the case E ≠ F to the transfer factors in the case E = F.

Let us first consider the split case in detail. Fix a split torus in M resp. M̂ as in [W1], together with a Borel subgroup. Consider the θ-stable torus T_{G} = Res_{E/F} (D), where D is the split E-torus of diagonal matrices in GSp(4, E). Consider its dual T̂_{G} with some embedding into Ĝ together with a Borel group B̂ as in [W1]. There is a map norm : (T_{G}^{s})_{θ} ≅ T_{M}^{s}, in the sense of [KS]. Namely one has (T̂_{G})θ̂ ≅ T̂_{M}^{s} asin [KS], and the

induced map norm : T_{G}(F) → T_{M}^{s}(F) is

norm : ( \begin{matrix}x_{1} & 0 & ∗ & ∗ \\ ∗ & x_{2} & ∗ & ∗ \\ 0 & 0 & x′_{1} & ∗ \\ 0 & 0 & 0 & x′_{2} \end{matrix} ) ↦ ( \begin{matrix}N_{E/F} (x_{2}) & ∗ \\ 0 & N_{E/F} (x_{1}) \end{matrix} ) × ( \begin{matrix}N_{E/F} (x′_{1}/x_{2}) & 0 \\ ∗ & 1 \end{matrix} ) ,

where x_{1}, x′_{1}, x_{2}, x′_{2} are in E, subject to the relation x_{1}x′_{1} = x_{2}x′_{2}. The stars indicate our choice of Borel subgroups.

The restricted root system R_{res}(G, T_{G}) ≅ R(G^{θ}, T_{G}^{θ}) of the diagonal torus T_{G} ≅ Res_{E/F} (E^{∗})^{3} restricted to T_{G}^{θ} ≅ (F^{∗})^{3} (diagonal matrices with entries in F^{∗}) can be identified with the root system of the group GSp(4, F). For these tori we can choose the a-data and χ-data as follows: χ_{λ} = 1 for all restricted roots and a_{λ} = 1 for the positive restricted roots and a_{λ} = −1 for the negative restricted roots. This gives the following transfer factors.

4. Transfer factors for the split torus in M:

The first factor Δ_{I}(γ, δ) is trivial, because this factor is of cohomological nature. It is attached to the Galois cohomology group M^{1}(F, T^{x}), where T^{x} is the intersection T_{sc}∩G^{x}. G^{x} is the group of fixed points of the lift θ_{sc} to G_{sc}, where G_{sc} is the simply connected covering of G_{der}. In our case G_{der} = Res_{E/F} Sp(4, E) is simply connected with fixed group G^{x} = Sp(4, F). Thus T^{x} ≅ (F^{∗})^{2} has vanishing cohomology by Hilbert 90.

The factor Δ_{II}(γ, δ) is also trivial. It is attached to the restricted roots, which do not come from M. It vanishes, because the χ-data are trivial.

The fourth factor ([KS], 4.5.A) is

Δ_{IV} (γ, δ) = ∏_{α_{res}} |Nα(δ) − 1|1/2 ; \overline{F} .

The product is over all restricted roots not from M. They are all of type R_{1}. Furthermore

Nα = ∑_{i=0}^{l_{α}−1} θ^{i}α .

Here l_{α} = n is the length of the Galois orbit of α. In our case Nα(δ^{∗}) = α_{res}(N_{θ}(δ^{∗})) = α_{res}(t), where t ∈ T_{θ}(F) is the image of γ ∈ T_{M}(F) under the admissable embedding T_{M} → T_{θ}. Observe, that δ^{∗} ∈ T (\overline{F}), such that N_{θ}(δ^{∗}) = t ∈ T_{θ}(F). Further observe δ^{∗} = gδθ(g)^{−1} for some g ∈ G_{sc}(\overline{F}), with the notations of [KS], (3.3.4). In our case m, defined in [KS] (3.1), is the identity.

The third factor Δ_{III}(γ, δ) = ⟨V, A⟩ is defined by pairing the following two elements

V = (V, D) = inv(γ, δ, \overline{γ}, \overline{δ}) ∈ M^{1}(F, U ⟶^{1−θ} S)

A = (A^{−1}, s_{U}) ∈ M^{1}(W_{F} , Ŝ ⟶^{1−θ̂} Û) .

See [KS], p.25-30, p.49-50 and p.79 ff. The general definition is complicated. In our case it considerably simplifies, because we can use the description given on [KS], p.49-50. Especially we can replace U by T_{sc} and S by T_{1} = T as explained in [KS], p. 49. (We are in the quasisplit case, and θ = θ^{∗}). Assume, that δ^{∗} = gm(δ)θ(g)^{−1} ∈ G(\overline{F}) with m = id : G → G^{∗} = G. Furthermore g ∈ G_{sc}(\overline{F}). Such that

N_{θ}(δ) = δθ(δ)...θ^{n−1}(δ) = γ .

Next observe ψ = id ([KS], (3.1) and u(σ) = 1([KS], (3.1)). Thus v(σ) = gu(σ)σ(g)^{−1} = gσ(g)^{−1} ([KS] (3.3.6)). This means, that V ∈ H^{1}(F, T_{sc} →^{(1−θ)π}T) is the image of a class [D] ∈ H^{0}(F, T) under the natural map j : H^{0}(F, T) → H^{1}(F, T_{sc} → T) ([KS], (A.3.13)). This follows from the long exact sequence attached to hypercohomology and H^{1}(F, T_{sc}) = H^{1}(F, T_{der}) = 1 (Hilbert 90 and Shapiro’s lemma). In other words, we can assume v(σ) = 1 by modifying g ∈ G_{sc}(\overline{F}) from the left by an appropriate element in T_{sc}(\overline{F}). After this modification [(v(σ), δ^{∗})] = [(1, δ^{∗})]. The hypercocycle condition implies σ(δ^{∗}) = δ^{∗}, δ^{∗} ∈ T (F). Therefore V = j([D]) is the image of D = δ^{∗} ∈ H^{0}(F, T). Furthermore δ^{∗} ∈ G(F) is θ-conjugate over F to δ ∈ G(F), because v(σ) = 1 implies g ∈ G_{sc}(F).

The element A is given by a pair (a_{T}^{−1}, s), where

1) s ∈ (T̂)_{ad} is the image of the element s ∈ T̂, determined by the endoscopic data, under the map T̂ → (T̂)_{ad}.

2) a_{T} is the one cocycle in Z^{1}(W_{F} , T̂) such that ξ_{T_{M}} = a_{T} · ξ_{1}, where ξ_{T_{M}} is defined

by

LT_{M} → ^{L}M = M → ^{L}G ;

The first map is defined by the χ-data, the second by the embedding ξ determined by the endoscopic data. The second map ξ_{1} is defined by

L(T_{θ}) → ^{L}((Ĝθ̂)^{0}) → ^{L}G ;

The first map is again defined by (the induced) χ-data, the group Ĝθ̂ is connected in our case with ^{L}((Ĝθ̂)^{0}) = GSp(4, C) × W_{F} and trivial action of the Weil group W_{F}, the second map is the inclusion map (diagonal embedding).

\underline{Assumption}: Let the endoscopic embedding ^{L}M → ^{L}G be unramified, further assume

s ∈ Z(M̂)^{Γ}.

This assumption is satisfied in our case. Consider a Frobenius element σ_{F} in the Weil

group. For this element ξ_{T_{M}} (1 × σ_{F}) = ξ(1 × σ_{F}) = s ◁ σ_{F}, whereas ξ_{1}(σ) = 1 · n(σ) × σ =

1 ◁ σ, since the χ-data are trivial such that n(σ_{T}) = 1 (in the notation of [KS] and [LS]). This gives

a_{T} (σ_{F}) = (1, ..., 1, diag(1, −1, 1, −1)) ∈ T̂_{G} .

It satisfies the compatibility equation

(1 − θ̂)a_{T} (σ_{F}) = s^{−1}σ_{F} (s) = (1, ..., 1, diag(1, −1, 1, −1), diag(1, −1, 1, −1))

and defines a hypercocycle.

The factor Δ_{III}(γ, δ) is obtained by the pairing of these two classes j([D]) and A. Using [KS], (A.3.13) we get

⟨j([D]), A⟩ = ⟨[D], î(A)⟩_{L}^{−1} ,

where ⟨., .⟩_{L}

⟨., .⟩_{L} : H^{0}(F, T) × H^{1}(W_{F} , T̂) → C^{∗}

is the Langlands pairing, and

î: H^{1}(W_{F} , T̂ → T̂_{sc}) → H^{1}(W_{F} , T̂)

is the natural map. In our case î(A) = î(a_{T}^{−1}, s) = [a_{T}^{−1}]. Therefore

Δ_{III}(γ, δ) = ⟨δ, a_{T} ⟩_{L} .

It remains to compute this number. Since E/F and therefore also the class of a_{T} is unramified, we may proceed following [B], 9.5 and 6.3. Let T_{d} be the maximal F-split torus in T = Res_{E/F} ((E^{∗})^{3}). The Langlands pairing defines an isomorphism H^{1}(W_{F} , T̂) ≅ Hom_{cont}(T (F), C^{∗}), which is contravariant functorial in T. It is shown in [B], that unramified classes can be evaluated by passage to T_{d} via the map induced by the inclusion T_{d} ↪ T. In other words, the restriction of the unramified character χ : T (F) → C^{∗}, corresponding to [a_{T}] ∈ H^{1}(W_{F} , T̂) ≅ Hom_{cont}(T (F), C^{∗}), to the subgroup T_{d}(F) corresponds to the cocycle, which maps σ to the image of a_{T} (σ) under the map T̂ = C^{∗} ⊗X^{∗}(T) → C^{∗} ⊗X^{∗}(T_{d}). In our case the natural map from X^{∗}(T) → X^{∗}(T_{d}) is given by (x_{1}, .., x_{n}) ↦ ∑ x_{i}. Thus the image of a_{T} (σ) is given by the product of all components

of (1, ..., 1, diag(1, −1, 1, −1)), which is diag(1, −1, 1, −1) ∈ T̂_{d}. This element corresponds to

the unramified character χ_{0} × χ_{0} × χ_{0} of T_{d}(F), which maps

diag(x, y, x^{−1}t, y^{−1}t) ∈ T_{d}(F) ⊂ GSp(4, F) = G^{θ}(F)

to

χ_{0}(xyt) ,

where χ_{0} is the unique nontrivial, unramified quadratic character of F^{∗}. In other words

Δ_{III}(γ, δ) = χ̃_{0} × χ̃_{0} × χ̃_{0}(δ) , δ ∈ Res_{E/F} (T_{d})(F) = T_{d}(E) ⊂ GSp(4, E) ,

where χ̃_{0} is the unique nontrivial quadratic unramified character of E^{∗}. In other words χ̃_{0}(x) = χ_{2n}(Norm_{E/F} (x)), where χ_{2n} is the unramified character of F^{∗}, such that χ_{2n}(π_{F}) is an n-th root of −1. This gives as a final result

Δ(γ, δ) = χ_{2n} × χ_{2n} ×_{2n} (γ) · ∏_{α} |α(γ) − 1|1/2 ; \overline{F} .

The product is over the restricted roots of T_{M}, which do not come from M.

Once more let me note, that the twisted endoscopic embedding does not give back our fixed endoscopic embedding in the case E = F, n = 1. It does this only up to equivalence. Let us interprete this in terms of the endoscopic lift between unramified representations. If

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) × ( \begin{matrix}α′ & 0 \\ 0 & β′ \end{matrix} ) × σ_{F} ∈ ^{L}M

is given, then its twisted basechange in ^{L}Res_{E/F} (GSp(4)) is

s · (g, g, ..., g) ◁ σ_{F}

with

g = ( \begin{matrix}α & 0 & 0 & 0 \\ 0 & β′ & 0 & 0 \\ 0 & 0 & β & 0 \\ 0 & 0 & 0 & α′ \end{matrix} ) × σ_{F} ∈ ^{L}T_{G} .

This value corresponds to an unramified character of the maximal split torus T_{d}(F) in T_{G}. Its Langlands dual is given by

( \begin{matrix}α^{n} & 0 & 0 & 0 \\ 0 & −(β′)^{n} & 0 & 0 \\ 0 & 0 & β^{n} & 0 \\ 0 & 0 & 0 & −(α′)^{n} \end{matrix} ) × σ

in the L-group of T_{d}. In other words, the isomorphism class of the spherical constituent of the unramified principal series

Ind( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )^{Gl(2,F)}(ξ, η) ⊗ Ind( \begin{matrix}∗ & 0 \\ ∗ & ∗ \end{matrix} )^{Gl(2,F)}(ξ′, η′)

of M(F) is lifted to the isomorphism class of the spherical constituent of the unramified principal series representations

χ_{1} × χ_{2} ◁ χ

where

χ_{1} = ((ξ/ξ′) ∘ Norm_{E/F}) · χ_{0} , χ_{2} = ((η/ξ′) ∘ Norm_{E/F}) · χ_{0} , χ = (ξ′ ∘ Norm_{E/F}) · χ_{0} .

Here χ_{0} denotes the unique nontrivial unramified quadratic character of F^{∗}. The characters are considered on the diagonal torus T_{d} with ’entries’ in F^{∗}. In the case E = F, n = 1 this differs from the ”untwisted” lift by the multiplication with χ_{0} ×χ_{0} ×χ_{0} or the change

α_{0} ↦ −α_{0}, α_{1} ↦ −α_{1} and α_{2} ↦ −α_{2}.

5. Spherical lift: Put M_{1} = GL(2, F)^{2}. Then M = M_{1}/D and M_{1} is a zextension of M. This defines a lift from the isomorphism classes of irreducible spherical representations of E_{un}(M_{1}(F), D) with central character trivial on D to the isomorphism classes of irreducible spherical representations of E_{un}(Res_{E/F} (GSp(4, E))

Π : E_{un}(M_{1}(F), D) → E_{un}(Res_{E/F} (GSp(4, E)) .

The map Π is surjective. Its fibres are finite of cardinality 2n^{3}. The data α, β, α′, β′ (modulo the Weyl group) can be recovered only up to multiplying α, β, α′, β′ by n-th roots of unities ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4} such that ζ_{1}ζ_{2} = ζ_{3}ζ_{4} and up to the interchange α, β ↦ ζα′, ζβ′,

α′, β′ ↦ ζα, ζβ. where ζ^{n} = −1.

\underline{Lemma}: Let ϕ_{n} be the spherical function on Res_{E/F} GSp(4, E) charactarized by the property trπ(ϕ_{n}) = α_{0} + α_{0}α_{1} + α_{0}α_{2} + α_{0}α_{1}α_{2} for irreducible spherical representations with Satake parameters α_{0}, α_{1}, α_{2}. Let f^{(n)} be the spherical function on Gl(2, F) characterized by the property trσ(f^{(n)}) = α^{n} + β^{n} for irreducible spherical representations of Gl(2, F) with Satake parameters α, β. Put ϕ̃_{n}^{M_{1}} = f^{(n)} ⊗1_{Gl(2,o_{F})} −1_{Gl(2,o_{F})} ⊗f^{(n)}. This is a spherical function on M_{1} = Gl(2, F)^{2}. Then ϕ_{n}^{M_{1}}(g) = ∫_{D} ϕ̃_{n}^{M}(gz)dz satisfies

trσ_{1} × σ_{2}(ϕ_{n}^{M_{1}}) = trΠ(σ_{1} × σ_{2})(ϕ_{n}) .

\underline{Proof}: If σ_{1}, σ_{2} are irreducible spherical representations of Gl(2, F), whose central characters coincide and which are given by the Satake parameters α, β resp. α′, β′; then trσ_{1} × σ_{2}(ϕ_{n}^{M_{1}}) = α^{n} + β^{n} − (α′)^{n} − (β′)^{n}. The lift Π(σ_{1} × σ_{2}) has Satake parameters determined by α_{0}α_{1} = α^{n}, α_{0}α_{1}α_{2} = −(β′)^{n}, α_{0}α_{2} = β^{n} and α_{0} = −(α′)^{n}. Therefore trace

Π(σ_{1} × σ_{2}) = α^{n} + β^{n} − (α′)^{n} − (β′)^{n}.

6.Matching of elementary functions

We now consider matching of orbital integrals. Let M = Gl(2, F)^{2}/F^{∗} be the endoscopic group considered. Fix a regular semisimple element γ = δ_{M} ∈ M(F) and consider the orbital integral

O_{γ}^{M}(f^{M}) = ∫_{M_{γ}(F)\M(F)} f^{M}(h^{−1}γh)dh .

By our assumptions the centralizer M_{γ} is a maximal torus containing γ. The orbital integrals on M are automatically stable. The orbital integrals are those considered in [KS], (5.5), because we are in the situation M = M_{1} ([KS],(2.1) and (2.2)) and z_{σ} = 1 ([KS], (3.2) and θ_{M} = Int(γ_{0}) = id ([KS], (5.4)).

Two functions f, f^{M} on G(F), resp. M(F) are said to be matching if

SO_{δ_{M} θ_{h}}(f^{M}) = ∑_{δ} Δ(δ_{M}, δ)O_{δθ}(f)

for all strongly G-regular δ_{M} ∈ M(F), where

O_{δθ} = ∫_{Cent_{θ}(δ,G(F))} f(g^{−1}δθ(g))dg .

The sum (which might be empty) is over representatives for the θ-conjugacy classes in G(F) of elements δ ∈ G(F), whose ’norm’ is δ_{M}. ([KS], (5.5.1)). This means, that δ lies in the \overline{F} conjugacy class of A_{M/G}(δ_{M}) ([KS],3.3.3). Here A_{M/G} is the the composite of Cl_{ss}(M) → T_{M}/Ω_{M} → (T_{G})_{θ}/Ω^{θ} and the inverse of the isomorphism Cl_{ss}(G, θ) → (T_{G})_{θ}/Ω^{θ} induced by the abstract norm map N_{T,θ} : T → T_{θ}.

In our case θ_{M} = id, γ_{0} = 1, z_{σ} = 1 ([KS], (5.4)).

Important examples for matching functions are the elementary functions. These are defined as follows: Let denote T_{M}^{−} and T_{G}^{−} the strictly contractive subsets (with respect to the chosen Borel subgroups) in the split torus T_{M} of M respectively in the θ-stable torus T_{G}, which is the restriction of scalars from E to F of the diagonal torus in GSp(4, E).

Then for δ = diag(x_{1}, x_{2}, x′_{1}, x′_{2}) ∈ T_{G}, with x_{i}, x′_{i} ∈ E^{∗} satisfies

δ ∈ T_{G}^{−} iff |x_{2}/x_{1}| < 1 and |x_{1}/x′_{1}| < 1.

The abstract norm is N_{T,θ} : T_{G} → (T_{G})_{θ}. In our case (T_{G})_{θ} ≅ (T_{G})^{θ}. The latter is identified with T_{M} through the choice of an admissable embedding. This gives a norm

map norm : T_{G} → T_{M}. It maps δ to the element diag(Norm_{E/F} (x_{2}), Norm_{E/F} (x_{1})) × diag(Norm_{E/F} (x′_{1}/x_{2}), 1). It follows, that δ ∈ T_{G}^{−} implies |Norm_{E/F} (x_{2})/Norm_{E/F} (x_{1})| < 1

and

|1/Norm_{E/F} (x′_{1}/x_{2})| = |Norm_{E/F} (x_{2}/x_{1}) · Norm_{E/F} (x_{1}/x′_{1})| < 1 .

Therefore

norm : T_{G}^{−} ⟶ T_{M}^{−} .

Furthermore

norm : (T_{G}^{−})^{#} ⟶ T_{M}^{−} ,

where # is the reflection in the Weyl group attached to

diag(x_{1}, x_{2}, x′_{1}, x′_{2})^{#} = diag(x′_{1}, x_{2}, x_{1}, x′_{2}) .

The elementary functions f_{δθ} and f_{t} are defined as the characteristic functions of the sets of elements k^{−1}δaθk respectively k^{−1}tak, where k ∈ K (K the subgroup of unimodular matrices) and where a ∈ ^{0}A is in the maximal compact subgroup of the torus T_{G} resp T_{M}. The twisted orbital integrals of f_{δθ} resp. f_{t} vanish for elements outside the (twisted) conjugacy classes of elements in T_{G} respectively T_{M} by definition. If norm(δ) = t, then there exists a constant c such that f_{δθ} and f_{t} are matching. Use [La], lemma 1, 2 and 3 to obtain, for a suitable normalisation of measures, \underline{matching} of elementary functions:

D_{G,θ}(δ)^{−1}f_{δθ}(x) and μ(Nm_{E/F} (δ))D_{M}(norm(δ))^{−1}f_{norm(δ)}(x) are matching.

Note μ(t) = χ_{0} × χ_{0} × χ_{0}.

\underline{Consequence}: For admissable representations π of finite length of M(F) one has, for suitably chosen measures,

T r(D_{M}(t)^{−1}f_{t}, π) = ∫_{^{0}T_{M}} D_{M}(ta)χ_{π}(ta)da
= δ_{B_{M}}^{−1/2}(t) ∫_{^{0}T_{M}} χ_{π(B)}(ta)da
= dimC(r_{BG}(π))^{^{0}T_{M}} (t) .

This follows from the Weyl integration formula, the Deligne-Casselman formula for characters and D_{M}(x) = δ_{B_{M}}^{−1/2}(x) for x ∈ T_{M}^{−}.

\underline{Twisted version}: For admissable θ-stable representations Π of finite length of G(F) one has, for suitable measures,

T r(D_{G,θ}(t)^{−1}f_{δθ} ◁ θ, Π) = ∫_{^{0}T_{G}} D_{G,θ}(δa)χ_{Π}(aδθ)da

= δ_{B_{G}}^{−1/2}(N_{E/F} (δ) ∫_{^{0}T_{G}} χ_{Π(B)}(aδθ)da

= dimC(r_{BG}(Π))^{^{0}T_{G}}(δθ) .

This follows from the twisted Weyl integration formula, the twisted character formula proved by Clozel, and the formula D_{G,θ}(δ) = δ_{B}(Norm_{E/F} (δ))^{−1/2}. The latter follows e.g. from [KS], (4.5), and the fact that all restricted roots are of type R_{1}.

The traces above are zero unless the representation π resp. Π has fixed vectors under the Iwahori group. Thus, if the representations are irreducible, they must be constituents of an unramified principal series representation Ind(λ), resp. Ind(λ̃) for some λ̃ = λ∘Norm_{E/F} (unitary normalized induction). This is explained in [La], p.526-527. One has the Shintani identity

r_{BG}(Ind(λ̃))(δθ) = r_{B′G′}(Ind(λ))(Norm_{E/F} (δ)) ∈ M(F)

where G′ = GSp(4, F) corresponds to the special case E = F.

Let us put things together

\underline{Lemma}: For unramified characters χ we get

T r(D_{M}(t)^{−1}f_{t}, Ind_{B_{M} (F)}^{M(F)}(χ)) = T r(μ(Norm_{E/F} (δ))D_{G,θ}(δθ)^{−1}f_{δθ}, Ind_{B_{G}(F)}^{G(F)}(μ(χ∘Norm_{E/F})))

where t = norm(δ) and where μ = χ_{0} × χ_{0} × χ_{0}.

7. Spherical Projectors:

Both groups M and G are of adjoint type. This means, following the argument in [La], that one can find a finite C-linear combination f of elementary functions f_{δθ} for every finite set of inequivalent irreducible representations Π of G(F) with a distinguished spherical irreducible representation Π_{0}, such that T rΠ(f ◁ θ) = 0 unless Π ≅ Π_{0}, and T rΠ_{0}(f ◁ θ) = 1. This is implicitly done in [La] via the construction of projectors relative to the maximal exponents in the Jaquet module. The projector has the form

f_{Π_{0}}(x) = ∑_{i} c_{i} · D_{Gθ}^{−1}(δ_{i})μ(Norm_{E/F} (δ_{i}))f_{δ_{i}θ}(x) .

Then, for every unramified spherical π_{0} which maps under the endoscopic lift to Π_{0}, the function

∑_{i} c_{i} · D_{M}^{−1}(norm(δ_{i}))f_{norm(δ_{i})}(x)

is a projector on the extremal exponent in the Jaquet module of π_{0}.

Assume Π_{0} is the Langlands quotient of I(λ̃) = Ind_{B(F)}^{G(F)}(λ̃). Then it is enough to construct a projector for the extremal exponent of r_{BG}(I(λ̃)). A projector onto the extremal exponent is #W_{G}(λ̃) times a projector onto the spherical constituent Π_{0} of the principal series representation I(λ̃). A similar relation holds for M. However, #W_{G}(λ) and #W_{M}(λ) need not be the same. Therefore we better consider

f_{Π_{0}}^{M}(x) = ∑_{i} c_{i}D_{M}^{−1}(norm(δ_{i}))\frac{1}{2}(f_{norm(δ_{i})}(x) + f_{norm(δ_{i})}(x^{#})) .

Observe that t^{#} is a norm of δ, if t is a norm. [KS], (3.3.1) and (3.3.3). Therefore this function f_{Π_{0}}^{M} has two properties

1) For irreducible π its traces are T rπ(f_{Π_{0}}^{M}) = 1 or 0, depending on whether π is spherical and lifts to Π_{0} or not (evaluated on a finite set of given representations, among only those constituents of an unramified I(λ) occur, for which the corresponding I(λ̃) was considered for the construction of f_{Π_{0}}).

2) The functions f_{Π_{0}} and f_{Π_{0}}^{M} are matching.

8. The fundamental lemma

The fundamental lemmas (for θ-semisimple and strongly θ-regular elements) in the twisted endoscopic setting for (M,^{L}M, s, ξ) and all relevant tori is now a consequence of the twisted trace formula, its stabilisation in [KS2] and the fundamental lemma for the unit element of the spherical Hecke algebras at almost all places. See [La] and [Cl]. The essential idea here stems from [K]. The argument carries over verbatim from [La]. We therefore skip the details. However it remains to establish the twisted fundamental lemma for the unit element for almost all places, and all relevant tori.

9. The fundamental lemma (unit element)

Put

O_{t}^{G′,κ}(f′) = ∑_{t′′} κ(inv(t, t′′))O_{t′′}^{G′}(f′)

SO_{t_{M}} (g) = ∑_{g′} O_{g′}^{M}(g) .

O_{δθ}^{G,κ}(f) = ∑_{δ′′} κ(inv(δ, δ′′))O_{δ′′θ}^{G}(f)

The sum are over representatives of (θ)-conjugacy classes in the stable (θ)-conjugacy class of t, t_{M} resp. δ. We assume that t_{M} ∈ M(F), t ∈ G′(F). Furthermore we assume δ ∈ G(F) = G′(E), such that N_{θ}(δ) is conjugate to t ∈ G′(F) in G′(\overline{F}). If such δ does not exist in G(F), then t is not a norm and we put O_{δθ}^{G,κ}(f) = 0.

By assumption there exist hyperspecial maximal compact subgroups K ⊂ G(F), K′ ⊂ G′(F) and K_{M} ⊂ M(F). We choose K by extension of scalars from K′.

The fundamental lemma for the unit element for nonarchimedian places, whose residue characteristic is different from 2, will be deduced from the corresponding unit-element fundamental lemmas for the basechange Res_{E/F} (GSp(E)), GSp(4, F) (see [Ko], prop.1) and the ordinary endoscopic unit-element fundamental lemma considered for the pair GSp(4, F), M(F) (see [W1]). Both together give

Δ(M, G′, t_{M}, t)Δ^{κ}(G′, G, t, δ)O_{δθ}^{G,κ}(1_{K}) = Δ(M, G′, t_{M}, t)O_{t}^{G′,κ}(1_{K′})
= SO_{t}^{M}(1_{K_{M}}) .

\underline{Assumptions}: Let us place ourselves in a more situation. Assume G′_{der} and G_{der} = Res_{E/F} G′_{der} to be simply connected. We assume, that ^{L}M → ^{L}G is unramified and that K = K_{G}, K′ = K′_{G′} satisfy the conditions of [Ko,unit]. Here t ∈ T_{G′}(F) is the image of t_{M} ∈ T_{M}(F) under the admissable embedding, and t = norm(δ^{∗}), δ^{∗} ∈ T_{G}(\overline{F}). Assume t to be regular. The factor Δ^{κ}(G′, G, t, δ) is not a transfer factor in the sense of [L-S]. It appears implicitely via the normalisations made in [Ko], prop.1, and need only to be defined if O_{δθ}^{G,κ}(1_{K}) ≠ 0. It order to proof the fundamental lemma for 1_{K}, where 1_{K} is the unit of the Hecke algebra, it therefore remains to check the coincidence of the transfer factors for elements δ ∈ G(F), stably θ-conjugate to δ^{∗}, which are in the support of O_{δθ}^{G,κ}(1_{K}). In other words

\underline{Lemma 1}: Δ(M, G, t_{M}, δ) = Δ(M, G′, t_{M}, t)Δ^{κ}(G′, G, t, δ) , if O_{δθ}^{G,κ}(1_{K}) ≠ 0.

\underline{Remark}: Observe Δ(M, G′, t_{M}, t) = Δ(M, G′, t_{M}, t) holds for elements t, which are Fconjugate to elements in K_{G′} = supp(1_{G′}). This follows from the fact that μ = χ_{0} ×χ_{0} ×χ_{0} is an unramified character.

Fix a unimodular matrix

M = ( \begin{matrix}b & a \\ 1 & n \end{matrix} ) ∈ Sl(2, ZZ)

where n = [E : F]. Using the notation of [Kottwitz, unity], we may consider t ∈ G′(F) and δ ∈ G′(E) with the property t ↔ δ (relative to M), as in [Ko,unity]: Let L be the completion of F^{un}, the maximal unramified extension of F. Before we explain this condition note that t and σ_{F} generate a subgroup in G′(L) × ⟨σ_{F} ⟩ isomorphic to ZZ^{2}. The same holds true for δσ_{F} and σ_{F}^{n}. We may ask whether these two subgroups ZZ^{2} are conjugate under some c ∈ G′(L); more precisely such that Int(c) ∘ M maps the generators to the generators

(A) Int(c)(t^{a}σ_{F}^{n}) = σ_{F}^{n}
(B) Int(c)(t^{b}σ_{F}) = δσ_{F} .

If this is the case, then one writes t ↔^{M} δ.

If M is fixed, we will only write t ↔ δ. For every choice of M, there exists an identity between between κ orbital integrals on G′(F) and twisted κ orbital integrals on G′(E) = G(F), as used above.

We first assume b = 1 and a = n − 1 (other choices are of type a′ ∈ a + nZZ). Using this correspondence the factor Δ^{κ}(G′, G, t, δ), for the elements δ in the support of O_{δθ}^{G,κ}(1_{K}) introduced above, can be determined from Kottwitz, loc. cit.. It is characterized by the following two properties

(i) Δ^{κ}(G′, G, t, δ) = 1 if t ↔ δ.
(ii) Δ^{κ}(G′, G, t′, δ′) = κ(inv(t′, t))Δ^{κ}(G′, G, t, δ)κ(inv(δ, δ′)).

For the second property remember the following: For a fixed maximal torus T in G,

one has D(T) = Ker(H^{1}(F, T) → H^{1}(F, G)). It is a subset of H^{1}(F, T). For [ξ] ∈ D(T) we

have ξ(σ) = A^{−1}A^{σ}, A ∈ G(\overline{F}). Then T_{ξ} = AT A^{−1} is a torus in G defined over F. The map ϕ = ϕ_{T_{ξ}T} : T → T_{ξ}, given by t ↦ AtA^{−1} is defined over F, i.e. ϕ(σ(t)) = σ(ϕ(t)). It defines an F-isomorphism between the tori T and T_{ξ}. Furthermore (ξ, A) define a class in H^{1}(F, T → G). Let [η] ∈ D(T_{ξ}) be a class with torus (T_{ξ})_{η} and map ϕ_{(T_{ξ})_{η}T_{ξ}} : T_{ξ} → (T_{ξ})_{η} defined by η(σ) = B^{−1}B^{σ}, B ∈ G(\overline{F}). Then the composite map ϕ_{(T_{ξ})_{η}T_{ξ}}ϕ_{T_{ξ}T} : T → (T_{ξ})_{η} maps t to BAtA^{−1}B^{−1}. This map is the map ϕ_{T_{ζ}T} attached to ζ(σ) = A^{−1}B^{−1}B^{σ}A^{σ} =

A^{−1}η(σ)Aξ(σ). Therefore (ϕ_{T_{ξ}T})_{∗}^{−1}(η)(σ) = A^{−1}η(σ)A implies

[ζ] = (ϕ_{T_{ξ}T})_{∗}^{−1}([η]) + [ξ] .

In other words

inv(T, T_{ζ}) = inv(T, T_{ξ}) + (ϕ_{T_{ξ}T})_{∗}^{−1}(inv(T_{ξ}, T_{ζ})) .

Similar one proves ϕ_{T_{ξ}T} (D(T)) = ϕ_{T_{ξ}T} (ξ) + D(T_{ξ}). However, in the p-adic case the sets D(T) are subgroups of H^{1}(F, T). This implies ϕ_{T_{ξ}T} (D(T)) = D(T_{ξ}). As we are in the p-adic case, we can therefore tacitly identify all the groups D(T_{ξ}) with one of them, say D(T), using the isomorphisms ϕ_{T_{ξ}T}. This allows to transfer characters κ on D(T) to characters κ_{ξ} on the D(T_{ξ}). Using this for κ = ⟨., s⟩ we get

κ(inv(T, T_{ζ})) = κ(inv(T, T_{ξ}))κ_{ξ}(inv(T_{ξ}, T_{ζ})) .

For regular stably conjugate elements one therefore gets the analoguous formula

κ(inv(γ, γ′′)) = κ(inv(γ, γ′))κ_{ξ}(inv(γ′, γ′′)) .

Therefore O_{γ}^{κ}(f) = κ(inv(γ, γ′))O_{γ′}^{κ}(f). This implies that Δ^{κ}(G′, G, t, δ) has to be consistant with the transformation property. That proves (ii).

One also needs another formula, which is necessary in order to complete the comparison with the Kottwitz formula for the traces of Frobenius ([Ko,Shimura varieties], (7.2) and (7.3)). This formula is valid not only if O_{δθ}^{G,κ}(1_{K}) ≠ 0, but at least also in case O_{δθ}^{G,κ}(ϕ_{n}) ≠ 0 , n = [E : F] ,, for the functions ϕ_{n} defined in Kottwitz [Shimura var.], p.173. See also p.180 formula (7.2) and (7.3). For our purposes it is enough to consider this formula in the setting of loc. cit. (7.2), i.e we make the assumptions that s ∈ Z(M̂)^{γ} and that the embedding ^{L}M → ^{L}G is unramified. Especially G is unramified over F. What we need is the comparison lemma

\underline{Lemma 2}: For the group G′ = GSp(4, F) the following assertion holds.

\underline{Assertion 2}: Suppose δ ∈ G(F) is such that O_{δθ}^{G,κ}(f) ≠ 0 either for f = 1_{K} or f = ϕ_{n}, n = [E : F]. Then

Δ(M, G, t_{M}, δ) = Δ(M, G′, t_{M}, t)⟨α(t; δ)), s⟩ ,

where κ(α(t; δ)) = ⟨α(t; δ), s⟩.

\underline{Definition of α(t; δ)}: Let us repeat the definition of

α(t; δ) ∈ X^{∗}(T̂_{G′}^{Γ}) , t ∈ T_{G′}(F) .

Suppose δ ∈ G(F) = G′(E). Then N_{θ}(δ) = N_{σ_{T}} (δ). Suppose that

N_{θ}(δ) = N_{σ_{F}} (δ) = δ · σ_{F} (δ)... · σ_{F}^{n−1}(δ) = dtd^{−1} , d ∈ G′(\overline{F}) .

By a theorem of Steinberg, therefore there exists an element c ∈ G′(L), where L is the completion of the maximal unramified extension F^{un} of F, such that

N_{σ_{F}} (δ) = δ · σ_{F} (δ)... · σ_{F}^{n−1}(δ) = ctc^{−1} , c ∈ G′(L) .

Then

b = c^{−1}δσ_{F} (c) ∈ T_{G′}(L) , (T_{G′} = G′_{t}) .

This element is uniquely defined up to σ_{F} -conjugacy in T_{G′}(L). Hence it determines a unique class [b] in B(T_{G′}) = H^{1}(⟨σ_{L}⟩, T_{G′}(L)). One has a canonical isomorphism ([Ko,iso])

ν : X_{∗}(T)_{Γ} → B(T) .

The inverse image of b under this isomorphism gives α(t; δ). The composition

T (F) → B(T) →^{ν^{−1}}X_{∗}(T)_{Γ}

is a homomorphism. It is not difficult to see, that its kernel K(T) is a subgroup of finite index in the maximal compact subgroup ^{0}T of T (F). This is clear for anisotropic tori S, because then B(S) is finite. In general consider a maximal split torus T_{d} in T. Then S = T/T_{d} is anisotropic. The claim is easily reduced to the split case by the exact sequence B(T_{d}) → B(T) → B(S) → 0. A character χ : T (F) → C^{∗} is called \underline{elementary}, if it is trivial on K(T). Every elementary character is continuous. We get a functorial group homorphism

T̂^{Γ} → Hom_{elm}(T (F), C^{∗}) ⊂ Hom_{cont}(T (F), C^{∗}) .

This being said it is clear, that α(t; δ) can be evaluated against s ∈ T̂_{G′}^{Γ}. Note

X^{∗}(T̂_{G′}^{Γ}) = X_{∗}(T_{G′})_{Γ}

and

⟨., .⟩X_{∗}(T_{G′})_{Γ} × T̂_{G′}^{Γ} = X_{∗}(T_{G′})_{Γ} × (C^{∗} ⊗ X^{∗}(T_{G′}))^{Γ} → C^{∗} .

This pairing gives rise to an isomorphism

T̂_{G′}^{Γ} ≅ Hom(X_{∗}((T_{G′})_{Γ}, C^{∗}) .

Namely for a Γ module with trivial Γ action, Hom(Y, A)^{Γ} is the set of homomorphisms

f : Y → A satisfying f(γ^{−1}y) = f(y), γ ∈ Γ. This is equivalent to f(I) = 0 for I = Ker(Y →

Y_{Γ}). But the homomorphisms f : Y_{Γ} → A are the homorphisms f : Y → A such that

f(I) = 0.

\underline{Explicit description of ν}: Choose a lift α ∈ X_{∗}(T_{G′}) of α(t; δ). Consider a splitting field K/L of the extension of T_{G′} to L. There is a natural map Norm_{K/L} : T_{G′}(K) → T_{G′}(L) induced from Nm_{K/L} : Res_{K/L}(T) → T. Then [Norm_{K/L}(α(π_{K}))] = [b]. This is the explicit description of α(t; δ), given in [Ko, Iso], 2.5.

\underline{Remark}: The construction of α(t; δ) ∈ B(T_{G′}) gives rise to another class

γ(t; δ) = [c^{−1}σ_{F}^{n} (c)] ∈ B(T) .

It is clear, that N_{σ_{F}} (δ) ∈ G′(E) implies c^{−1}σ_{F}^{n} (c) ∈ T_{G′}(L). We have

N_{σ_{F}} (b) = c^{−1}N_{σ_{F}} (δ)σ_{F}^{n} (c) = c^{−1}σ_{F}^{n} (c)σ_{F}^{n} (c^{−1}N_{σ_{F}} (δ)c) = c^{−1}σ_{F}^{n} (c)σ_{F}^{n} (t) = c^{−1}σ_{F}^{n} (c) .

Since σ_{F} acts trivially on B(T_{G′}), we get

α(t; δ)^{n} = γ(t; δ)[t] ∈ B(T_{G′}) .

Before we discuss assertion 2 we show that lemma 1 is a consequence of assertion 2.

\underline{Reduction: assertion 2 implies lemma 1}: It is enough, that ⟨α(t; δ), s⟩ and Δ^{κ}(G′, G, t, δ)

coincide for elements δ with O_{δθ}^{G,κ}(1_{K}) ≠ 0. This will follow from the two properties

(i) κ(α(t; δ)) = 1 for t ↔ δ and O_{δθ}^{G,κ}(1_{K}) ≠ 0.
(ii) ⟨α(t′; δ′), s⟩ = ⟨inv(t′, t), s⟩⟨α(t; δ), s⟩⟨inv(δ, δ′)), s⟩.

Proof of (i) and (ii): The second property is a variant of [Ko, Shim], p.168 and p.170.

E.g. suppose t′ ∈ T_{ξ}, t ∈ T with T_{ξ} = AT A^{−1}, t′ = AT A^{−1}. Then b′ = (cA^{−1})^{−1}δσ_{F} (cA^{−1}) ∈ T_{ξ}(L), where A can be chosen in G′(L). Thus b′ = ϕ_{T_{ξ}T} (b)Aσ_{F} (A^{−1}) = ϕ_{T_{ξ}T} (b)inv(T_{ξ}, T), and hence α(t′; δ) = (ϕ_{T_{ξ}T})_{∗}(α(t; δ))inv(T_{ξ}, T) because inv(T_{ξ}, T) = −ϕ_{T_{ξ}T} (inv(T, T_{ξ})) =

[Aσ_{F} (A^{−1})]. Similar one deals with a change of δ.

The first property follows from the equations [Ko, unit] subject to j = 1 and our choice b = 1, a = n − 1. Namely Kottwitz shows that (A) and (B) are equivalent to (A’),(B’) or (C’),(D’) where

(A′) t^{n−1} = c^{−1}σ_{F}^{n} (c)
(B′) t = c^{−1}δσ_{F} (c)
(C′) N_{σ_{F}} (δ) = δ · σ_{F} (δ)... · σ_{F}^{n−1}(δ) = ctc^{−1}
(D′) (δσ_{F})^{−a}σ_{F}^{a} = cσ_{F} (c)^{−1} .

\underline{Lemma 3}: Suppose t ∈ G′(F) and δ ∈ G′(E) are given. Then t ↔ δ (with respect to a = n − 1, b = 1) are equivalent to (B’),(C’) for some c ∈ G′(L).

\underline{Proof}: Reserve the notation t ↔ δ temporarily for a = n − 1, b = 1. Then, as already remarked (A),(B) implies (B’),(C’). Suppose, that conversely (B’),(C’) would imply (A’).

Then we are done, because (A),(B) and (A’),(B’) are equvalent. Under our assumptions N = N_{σ_{F}} (δ) ∈ G′(E). (B’) implies

t^{n} = N_{σ_{F}} (t) = c^{−1}N_{σ_{F}} (δ)σ_{F}^{n} (c) = c^{−1}σ_{F}^{n} (c)σ_{F}^{n} (c^{−1}Nc) .

This is equal to c^{−1}σ_{F}^{n} (c)t using (C’). Division by t gives (A’). The lemma is proved.

\underline{Corollary}: For t ∈ T′(F), δ ∈ G′(E) satisfying (C’), the conditions

1) α(t; δ) = [t] ∈ B(T′)
2) t ↔ δ (with respect to a = n − 1, b = 1)

are equivalent.

\underline{Proof}: If t ↔ δ then condition (B’) implies b = t, hence α(t; δ) = [t]. Suppose conversely α(t; δ) = [t] holds. Then we must show, that (B’) can be satisfied. By lemma 3 this implies t ↔ δ. If (B’) is not satisfied, we use our freedom in the choice of c. We are free to make a change c ↦ cc_{0}, c_{0} ∈ T′(L). Therefore one can read (B’) as [t] = α(t; δ) in the quotient B(T′). As this assumed for a suitable choice of c_{0} condition (B’) can be satisfied.

\underline{Lemma 4}: a) For t ∈ G′(F) there exists a δ ∈ G′(E) with t →^{M} δ iff (A’) can be solved for c ∈ G′(L). Any such δ is unique up to σ_{F} -conjugacy in G′(E). b) For δ ∈ G′(E) there exists a t ∈ G′(F) such that t →^{M} δ iff (D’) can be solved. Any such t is unique up to conjugacy in G′(F).

\underline{Proof}: [Ko,unity], p.241-242.

Now let us continue the proof of (i): Equation (B’) and (C’) imply α(t; δ) = [t] ∈ B(T_{G′}) according to lemma 3 and the corrollary thereafter. But O_{t}^{G′,κ}(1′_{G}) ≠ 0 follows from O_{δθ}^{G,κ}(1_{G}) ≠ 0 ([Ko,unity] prop.1, because we have chosen K and K′ subject to the conditions in [Ko,unity]. For the proof of (i) it therefore remains to show

\underline{Lemma 5}: The condition O_{t}^{G′,κ}(1_{K′}) ≠ 0 implies κ([t]) = 1.

\underline{Remark}: Lemma 5 strengthens the following fact: The condition O_{t}^{G′}(1_{K′}) ≠ 0 implies [t] ∈ Ker(B(T′)_{tor} → B(G′)) (uses functoriality of B(.)), which is canonically isomorphic to D(T′) = Ker(H^{1}(F, T′) → H^{1}(F, G′)). Namely it exists an integer k such that t^{k} = as, where a is in a maximal split subtorus A of T′ and s in a anisotropic subtorus S of T′. See [Casselman]. The eigenvalue conditions imply a ∈ ^{0}A, therefore the image of a in B(A) is zero (use the explicit description of map A(F) → B(A). On the other hand B(S) = X_{∗}(S)_{Γ} is torsion. Therefore [t] ∈ B(T′)_{tor}. The rest follows from the fact, that t is conjugate to an element in K′ and [Ko,unit], 1.(b) and

\underline{Lemma 6}: Let G be unramified, let λ_{G} be the map from [Milne, B.16]. It is surjective, every hyperspecial subgroup is in the kernel. The diagram

G(F) →  π_{1}(G)^{Γ}
↓  ↓
B(G) → π_{1}(G)_{Γ} = X^{∗}(Ẑ(G)^{Γ})

is commutative. Note X^{∗}(Ẑ(G)^{Γ}) = X^{∗}(Ẑ(G))_{Γ} and π_{1}(G) ≅ X^{∗}(Z(Ĝ)) ([Milne,B.28]).

\underline{Proof}: The lower horizontal map is normalized as in Kottwitz and is the inverse of [Milne], B.26-27. By [Milne], B.10, B.16 and B.26 the proof is easily reduced to the case, where G is an unramified torus over F. This case is easily reduced by [Milne] (B.15.1) to the situation, where F = L and where G is a split torus over L.

\underline{Proof of lemma 5}: From [Ko,unit], prop. 1 follows that O_{δθ}^{G}(1_{K}) ≠ 0 implies O_{δθ}^{G,κ}(1_{K}) = O_{t}^{G′,κ}(1_{K′}), provided δ ↔ t holds. Here

O_{δθ}^{G,κ}(1_{K}) = ∑_{δ′′} κ(inv(δ, δ′′))O_{δ′′θ}^{G}(1_{K})

O_{t}^{G,κ}(1_{K′}) = ∑_{t′′} κ(inv(t, t′′))O_{t′′}^{G′}(1_{K′}) .

Let us assume δ ↔ t, with respect to the parameters a = n − 1, b = 1. According to [Ko,unit], one can find δ ↔ t′ for a modified choice of the parameter a′ = a + kn, such

that inv(t, t′) = [t^{−k}] ∈ B(T′)_{tor} = H^{1}(F, T′). Choose k = −1. Then [t] = inv(t, t′).

From prop.1, loc.cit it follows that

O_{t}^{G′,κ}(1_{K′}) = O_{δθ}^{G,κ}(1_{K}) = O_{t′}^{G′,κ′}(1_{K′}) ≠ 0 .

But κ(inv(t, t′′)) = κ(inv(t, t′))κ′(inv(t′, t′′)), where κ′∘ϕ = κ and ϕ_{G′_{t′} G′_{t}} : G′_{t} ≅ G′_{t′}. Therefore

O_{t}^{G′,κ}(1_{K′}) = κ(inv(t, t′))O_{t′}^{G′,κ}(1_{K′}) ≠ 0 .

This implies κ(inv(t, t′)) = κ([t]) = 1. Lemma 5 is proved.

\underline{Remark}: If the fundamental lemma holds for the pair 1_{K_{M}} , 1_{K′}, this implies O_{t}^{G′,κ}(1_{K′}) ≠ 0 for all t ∈ ψ(T_{M} ∩ K_{M}). Note, ψ : T_{M} ≅ T′ denotes the admissable embedding. Thus ψ(T_{M} ∩ K_{M}) is contained in the kernel of the character

κ ∘ [.] : T′ ∩ K′ → D(T′) → C^{∗} .

This follows from lemma 3. Is it equal to the kernel? In other words, does κ([t]) = 1 for

t ∈ T′ ∩ K′ imply O_{t}^{G′,κ}(1′_{K}) ≠ 0?

11.The norm map and stably conjugacy:

Let F be a local nonarchimedian field of characteristic zero and let E/F be an \underline{unramified} cyclic field extension of degree n. Let G be the Weil restriction Res_{E/F} (G′/E) of the reductive group G′ over F. Assume that G′ is \underline{quasisplit} over F with Borelgroup B′ and maximal torus T′_{0} in B′, defined over F. The Frobenius automorphism σ_{F}, a generator of the Galois group of E/F, acts by an algebraic F-automorphism θ on G. We have (G, θ) as in the last section. Then we are in the situation of [KS], where one can study twisted elliptic endoscopic groups attached to (G, θ).

In [KS] several definitions are necessary. In our special case, most of these definitions turn out to be either rather easy or unneccessary.

First of all, the groups G′ and G are quasisplit; B = Res_{E/F} (B′) and T_{0} = Res_{E/F} (T′_{0}) is a Borel group in G. The automorphism θ respects a pair (B, T_{0}). It even preserves a suitable F-splitting! This implies G = G^{∗}, θ = θ^{∗} and ψ = id in the sense of [KS], (1.2). The fixed group T_{0}^{θ} is isomorphic to T_{0}. It defines a maximal F-split torus T_{d} in G.

Fix some \underline{unramified} endoscopic datum (^{L}M, M, s, ξ) for the group G′. Furthermore

\underline{assume} s ∈ Z(M̂)^{Γ}.

Let ψ : T_{M} ↪ T′ ⊂ G′ be an admissable embedding attached to the given endoscopic datum. Put T = Res_{E/F} (T′). It is a maximal torus in G. We saw, that there is a canonical isomorphism T_{θ} ≅ T′, which defines the abstract norm map Nm_{E/F} : T → T′.

Fix some γ_{M} ∈ T_{M}(F). Let t denote its image in T′(F), γ (or γ_{0}) its image in T_{θ}(F). We choose some element δ^{∗} ∈ T (\overline{F}), such that

Nm_{E/F} (δ^{∗}) = t ∈ T′(F) .

Always suppose, that t is \underline{regular}. Then we can consider the finitely many F-conjugacy classes of elements in G′(F), which are stably conjugate to t. Choose representatives

t = t_{1}, t_{2}, .... ∈ G′(F) , inv(t, t_{i}) ∈ D(G, T′) .

Furthermore consider the finitely many θ-conjugacy classes in G(F) of elements in G(F), which are stably θ-conjugate to δ^{∗}. This set may be empty. If not, choose representatives

δ_{1}, δ_{2}, ... ∈ G(F) .

Note, that δ^{∗} need not be conjugate to these representatives in G(F). However, we have

δ^{∗} = g_{i}δ_{i}θ(g_{i})^{−1} , g_{i} ∈ G_{sc}(\overline{F}) .

Attached to these data, are the cocycles

v_{i}(σ) = g_{i}σ(g_{i})^{−1} ∈ T_{sc}(\overline{F}) .

Obviously

σ(δ^{∗}) = v_{i}(σ)^{−1}δ^{∗}θ(v_{i}(σ))^{−1}

holds. Abbreviate (1−θ)T = V and let i : V → T be the inclusion map. Since σ(δ^{∗}) = σ mod

V = (1 − θ)T, we can find λ ∈ T (\overline{F}), such that σ(δ^{∗}) = σλ^{1−θ}. Then (v_{i}(σ)λ)^{−1}δ^{∗}θ(v_{i}(σ)λ) =

σ(δ^{∗}). This implies v_{i}(σ)λ ∈ T^{θ}(\overline{F}). Therefore v_{i}(σ) ∈ T (\overline{F}) and

δ^{∗}σ(δ^{∗})^{−1} = v_{i}(σ)^{1−θ} .

Let π : T_{sc} → T be the natural map. The pair (v_{i}(σ), δ^{∗}) defines a hypercocycle in Z^{1}(Γ_{F} , T_{sc} ⟶^{(1−θ)π} T). Consider its class

V_{i} = [(v_{i}(σ), δ^{∗})] ∈ H^{1}(F, T_{sc} ⟶^{(1−θ)π} T) ,

defined in [KS], p.49 (remember that T_{1} = T holds in our case, using the notations of loc. cit.). It is independent of the choice of the chosen δ^{∗} ∈ T (\overline{F}) such that Nm_{E/F} (δ^{∗}) = t, because any other choice is given by δ^{∗}w^{1−θ}, w ∈ T (\overline{F}). This change gives rise to a hypercoboundary.

Furthermore consider the class

inv_{θ}(t, δ_{i}) = [(v_{i}(σ), δ^{∗})] ∈ H^{1}(F, T ⟶^{(1−θ)} T) .

It is the image of V_{i} under the map induced from the complex map (π, id)

\begin{matrix}T_{sc} & →^{(1−θ)π} & T \\ ↓ π & & ↓ id \\ T & →^{(1−θ)π} & T \end{matrix} .

Note T^{θ} = T′, T_{θ} ≅ T′ and

0 → V → T →^{Nm_{E/F}} T′ → 0

0 → T′ → T →^{(1−θ)} V → 0 .

\underline{Special case}: Suppose that δ^{∗} can already be chosen in T (F), not only in T (\overline{F}). Then there are also the invariants

inv(δ^{∗}, δ_{i}) ∈ Ker(H^{1}(T, T^{θ}) → H^{1}(F, G)) = D(T, θ) ,

which parametrize conjugacy classes within a stable conjugacy class. Also (v_{i}(σ), δ^{∗}) is a product of two hypercocycles (v_{i}(σ), 1)(1, δ^{∗}). The class of the first is the image of

inv(δ^{∗}, δ_{i}) under the map (T →^{1−θ} T) → T^{θ}

\begin{matrix}T & →^{1−θ} & T \\ ↑ id & & ↑ incl \\ T & →^{1−θ} & V \\ ↑ incl & & ↑ 0 \\ T^{θ} & → & 0 \end{matrix} .

In the derived category the second map is a quasiisomorphism. With this morphism as the first map one gets a distinguished triangle

(T^{θ} → 0) → (T →^{1−θ} T) → (0 → T_{θ}) →

because T_{θ} = T/V. In our case both T^{θ} and T_{θ} can be identified with T′.

There a second distinguished triangle attached to

(0 → T) → (T →^{1−θ} T) → (T → 0) → ,

the maps being (0, id) and (id, 0).

We get from the two distinguished triangles above

\begin{aligned}& & H^{0}(F, T) \\ & & ↓ \\ & & H^{0}(F, T) \\ & & ↓ & ↘ Nm \\ 0 → H^{1}(F, T^{θ}) & → & H^{1}(F, T →^{1−θ} T) & → & H^{0}(F, T_{θ}) → H^{2}(F, T^{θ}) \\ & r ↘ & ↓ \\ & & H^{1}(F, T) \\ & & ↓ \\ & & H^{1}(F, T) \end{aligned} .

The maps r and Nm, making the triangles in the diagram commutative, are induced from the norm Nm_{T,θ} : T → T_{θ} and from the inclusion T^{θ} ↪ T. In our case we may identify T^{θ} and T_{θ} with T′. Then these maps are the norm map

Nm = Norm_{E/F} : T (F) = T′(E) → T′(F)

and the map

r = res : H^{1}(F, T′) → H^{1}(E, T′) .

Observe that H^{1}(F, T) ≅ H^{1}(E, T′) by Shapiro’s lemma. This isomorphism is induced by the compositionof two maps, the restriction map H^{1}(F, T) → H^{1}(E, T/E) and the projection on the one component given by T/E = ∏ T′. Because the restriction map is functorial, the map induced from the inklusion T′ = T^{θ} ↪ factors over the restriction map H^{1}(F, T′) → H^{1}(E, T′). The inclusion T′(E) → T (E) = ∏_{i=1}^{n} T′(E) composed with the projection on the one component is the identity map. This determines r as claimed above.

Let ^{D} denote the Pontrajagin dual. The Tate-Nakayama isomorphism gives H^{1}(F, T) ≅ π_{0}(T̂^{γ})^{D}, in a reformulation due to Kottwitz. Therefore

Kokern(r)^{D} ≅ Kern(N : (π_{0}(T̂^{Γ_{F}})^{θ} → π_{0}((T̂′)^{Γ_{F}}) .

The map is induced by N : T̂ = ∏_{i=1}^{n} T̂′ → T′ by summation of components. Note

T̂^{Γ_{E}} = ∏_{i=1}^{n}(T̂′)^{Γ_{E}} and (T̂′)^{Γ_{F}} is the subset of all (x, σ_{F} (x), .., σ_{F}^{n−1}(x)), x ∈ (T̂′)^{Γ_{E}}.

Put Y = (T̂′)^{Γ_{E}}. Y and the connected component Y^{0} are preserved by the action of σ_{F} and one has an induced action on π_{0}(Y) = Y/Y^{0}. Therefore

π_{0}((Y^{r})^{σ_{F}}) ≅ {(c, σ_{F} (c), .., σ_{F}^{n−1}(c)) | c ∈ π_{0}(Y)} .

Therefore

π_{0}((Y^{r})^{σ_{F}})^{θ} ≅ {(c, c, ..., c) | c ∈ π_{0}(Y)^{σ_{F}} .

We get

Kokern(r)^{D} ≅ Kern(n : (π_{0}(T̂^{Γ_{E}})^{σ_{F}} → π_{0}((T̂′)^{Γ_{F}})

induced by the map N : T̂^{Γ_{E}} → T̂^{Γ_{F}}, which map x to ∑_{i=1}^{n} σ_{F}^{i} (x). Note that elements in c ∈ ((T̂′)^{Γ_{E}})^{σ_{F}} have representatives y ∈ Y = (T̂′)^{Γ_{E}} with

σ_{F} (y) = y + y^{0} , y^{0} ∈ Y^{0} .

The map N maps connected components to connected components. Therefore N(c) has representative

N(y + ((T̂′)^{Γ_{E}})^{0}) = ny mod ((T̂′)^{Γ_{F}})^{0} .

\underline{Example}: Let N_{L}^{1} be the one dimensional torus of elements of norm one in a quadratic extension field L/F. Let E be a quadratic field linear disjoint from from L/F. Then

(LE)^{∗} →^{N_{LE/E}} E^{∗} → H^{1}(E, N_{L}^{1}) → 1

L^{∗} →^{N_{L/F}} F^{∗} → H^{1}(F, N_{L}^{1}) → 1 .

We have H^{1}(E, N_{L}^{1}) ≅ ZZ/2ZZ and H^{1}(E, N_{L}^{1}) ≅ ZZ/2ZZ, but the restriction map between these two groups is trivial. Either use the desciption above, or alternatively the fact F^{∗} ⊂ N_{LE/E}((LE)^{∗}) ⊂ E^{∗}. More generally consider cyclic extensions E/F linear disjoint from L/F. Then the restriction map is trivial if n = [E : F] is even, and is an isomorphism if n = [E : F] is odd.

\underline{Digression} (probably nonsense): Let denote \overline{δ}^{∗} = pr_{1}(δ^{∗}) ⊂ G′(\overline{F}), for any elements δ^{∗} ∈ G(\overline{F}). Here pr_{1} denotes the projection onto the 1-component with respect to G(\overline{F}) = ∏_{i=1}^{n} G′(\overline{F}). The equation Nm_{E/F} (δ^{∗}) = t implies

\overline{N_{θ}(δ^{∗})} = N_{σ_{F}} (\overline{δ}^{∗}) = \overline{δ}θ(\overline{δ})....θ^{n−1}(\overline{δ}) = t .

But

\overline{δ}^{∗} = \overline{g}_{i}\overline{δ}_{i}θ(\overline{g}_{i})^{−1}

implies

t = \overline{N_{θ}(δ^{∗})} = \overline{g}_{i}N_{σ_{F}} (\overline{δ}_{i})\overline{g}_{i}^{−1} ,

because θ^{n}(\overline{g}_{i}) = \overline{g}_{i} ∈ G′(\overline{F}). Therefore

N_{σ_{F}} (\overline{δ}_{i}) = (\overline{g}_{i})^{−1}t\overline{g}_{i} .

Furthermore N_{σ_{F}} (\overline{δ}_{i}) ∈ G′(E). Its Galois behaviour is therefore completely determined by knowing the action of σ_{F}. But

= σ_{F} (N_{σ_{F}} (\overline{δ}_{i})) = (\overline{δ}_{i})^{−1}N_{σ_{F}} (\overline{δ}_{i})σ_{F}^{n} (\overline{δ}_{i}) = (\overline{δ}_{i})^{−1}N_{σ_{F}} (\overline{δ}_{i})δ_{i} .

This gives

σ_{F} ((\overline{g}_{i})^{−1}t\overline{g}_{i}) = (\overline{δ}_{i})^{−1}((\overline{g}_{i})^{−1}t\overline{g}_{i})\overline{δ}_{i} .

Therefore

\overline{g}_{i}\overline{δ}_{i}σ_{F} (\overline{g}_{i})^{−1} ∈ T′(\overline{F}) .

Furthermore

Image(\overline{v}_{i}(σ)) = \overline{g}_{i}σ(\overline{g}_{i})^{−1} ∈ T′(\overline{F}) , σ ∈ Γ_{E}

defines a 1-cocycle in Z^{1}(Γ_{E}, T′) over E!

The restricted root system R_{res}(G, T) can be identified with the root system R(G^{θ}, T^{θ}) = R(G′, T′). Especially it is reduced, all its roots are of type R_{1} in the sense of [KS], (1.3).

The dual group Ĝ is ∏_{i=1}^{n} Ĝ′, the Galois group acts as specified (using Ψ). Especially the Frobenius acts like the dual automorphism

σ_{F} (x_{1}, ..., x_{n}) = (σ_{F} (x_{2}), ..., σ_{F} (x_{n}), σ_{F} (x_{1})) .

Furthermore

θ̂(x_{1}, ..., x_{n}) = (x_{2}, ..., x_{n}, x_{1}) .

Let us consider the endoscopic datum (M,^{L}M, s, ξ) of G′. This gives rise to a twisted endoscopic datum (^{L}M, M, s_{E}, ξ_{E}) for G = G_{E}. In the case E = F the new datum will not be the same, but only equivalent to the original one. In order to define the new embedding, remember ξ was unramified. Let σ_{F} be the Frobenius automorphism in Gal(E/F). Then the new embedding ξ_{E} : ^{L}M → ^{L}G is defined by

ξ_{E}(1 × σ_{F}) = s_{E} × ξ(σ_{F}) = (1, ..., 1, s) × ξ(σ_{F}) .
ξ_{E}(h × 1) = (ξ(h), ξ(h), .., ξ(h)) × 1 .

It is clear that Cent(Ĝ, Int(s_{E} ∘ θ)) is Cent(G′, Int(s)), diagonally embedded. Therefore

ξ_{E}(M̂) = Cent(Ĝ, Int(s_{E} ∘ θ))^{0}. Furthermore Int(s_{E}) ∘ θ ∘ ξ_{E} = a′_{E} · ξ_{E}, by our assumption s ∈ Z(M̂)^{Γ}. Here a′_{E} = (1, ..., a′, 1), where Int(s)∘ξ = a′ ·ξ. We will \underline{assume} a′ = 1, therefore a′_{E} = 1.

For the tori T′ of G′, the tori T = Res_{E/F} (T′) are θ-stable. The maps T → T_{θ} ≅ T′ ≅ T_{M} are considered as explained above. They define admissable isomorphism. We choose a-data and χ-data according to our identifications R_{res}(G, T) = R(G^{θ}, T^{θ}) = R(G′, T′) to be the same for the pairs T_{M}, T′ and T_{M}, T. This gives the following transfer factors.

\underline{12. Comparison of transfer factors}:

Note, that G is quasisplit over F and θ fixes a F-splitting (B, T_{0}, {x_{α}}) of G. Therefore ψ = id, u(σ) = 1, m = id, z = 1 in the notation of [KS]. Furthermore G^{θ} = G′ and T^{θ} = T′ and T_{θ} ≅ T′ (not induced by the natural map T^{θ} → T_{θ}).

1) The first factor Δ_{I}(γ_{M}, δ) is of cohomological nature. It is attached to a class λ_{{a_{α}}}(T^{x}) in the Galois cohomology group H^{1}(F, T^{x}), where T^{x} is the intersection T_{sc} ∩ G^{x}. G^{x} is the group of fixed points of the lift θ_{sc} to G_{sc}, where G_{sc} is the simply connected covering of G_{der}. It is defined by the induced a-data via R(G^{x}, T^{x}) ⊂ R_{res}(G, T) (see [KS], (4.2)). But G^{x} = G_{sc}^{θ_{sc}} = G′_{sc} and T^{x} = T_{sc}^{θ_{sc}} = T′_{sc}. So the group T^{x} and the class λ_{{a_{α}}}(T^{x}) do not depend on E, and for E = F it is the same class as for the original endoscopic setting we started from.

The factor Δ_{I}(γ_{M}, δ) is the value of the Tate-Nakayama pairing of this class against a class s_{T,θ} in π_{0}((T̂^{x})^{Γ}) constructed from s_{E} resp. s_{F} or s. It is defined to be the image of s_{E} ∈ T̂_{M} ⊂ T̂_{G} under the maps T̂_{G} → T̂_{G_{ad}} → (T̂_{G_{ad}})_{θ_{ad}} → π_{0}((T̂^{x})^{Γ}). Note T̂^{x} = ˆ (T_{sc}^{θ_{sc}}) ≅ ((T̂)_{ad})_{θ_{ad}}. In our case the map T̂_{G_{ad}} → (T̂_{G_{ad}})_{θ_{ad}} maps the class of s_{E} = (1, ..., s) to the image of s = s_{F} in T̂_{G′_{ad}}. Therefore again the class in π_{0}((T̂^{x})^{Γ}) does not depend on E. For E = F it is the same as for the original endoscopic datum we started from. Therefore

Δ_{I}^{E}(γ_{M}, δ) = Δ_{I}^{F} (γ_{M}, t) = Δ_{I}(γ_{M}, t) .

We get the same conclusion for the factors Δ_{II} and Δ_{IV}. In all three cases, they are the same. Remember all roots are of type R_{1}. Therefore

Δ_{II}^{E}(γ_{M}, δ) = ∏_{α_{res}} χ_{α_{res}}(\frac{Nα(δ^{∗}) − 1}{a_{α_{res}}}) .

The sum is over the Galois orbits of restricted roots, which do not come from M. See [KS], lemma 4.3.A. Furthermore

Nα = ∑_{i=0}^{l_{α}−1} θ^{i}α .

Here l_{α} = n is the length of the Galois orbit of α. In our case Nα(δ^{∗}) = α_{res}(N_{θ}(δ^{∗})) = α_{res}(t), where t ∈ T′(F) ≅ T_{θ}(F) is the image of γ_{M} ∈ T_{M}(F) under the admissable embedding T_{M} → T_{θ}. Observe δ^{∗} ∈ T (\overline{F}), has norm \overline{N_{θ}(δ^{∗})} = t ∈ T′(F). See [KS], (3.3.4). In our case m, defined in [KS] (3.1), is the identity. We get

Δ_{II}(t, γ_{M}) = Δ_{II}^{F}(t, γ_{M}) = Δ_{II}^{E}(δ, γ_{M}) = ∏_{α} χ_{α}(\frac{α(t) − 1}{a_{α}}) .

The product is over the Galois orbits of roots in R(G′, T′), which do not come from M. This is obviously independent from E.

The fourth factor ([KS], 4.5.A) is treated similar:

Δ_{IV}^{E} (γ_{M}, δ) = ∏_{α_{res}} |Nα(δ) − 1|1/2 ; \overline{F} .

The product is over all restricted roots not from M. They are of type R_{1}. The same argument as above gives

Δ_{IV} (γ_{M}, t) = Δ_{IV}^{F} (γ_{M}, t) = Δ_{IV}^{E} (γ_{M}, δ) = ∏_{α} |α(t) − 1|1/2 ; \overline{F} .

The product is over the Galois orbits of roots in R(G′, T′), which do not come from M. It is obviously independent from the choice of E.

The third factor Δ_{III}(t, δ) = ⟨V, A⟩ is defined by the value of the pairing of the following two elements

V = [(v(σ), δ^{∗})] ∈ H^{1}(F, T_{sc} ⟶^{(1−θ)π} T)

A = (A^{−1}, s) ∈ H^{1}(W_{F} , T̂ ⟶^{π̂(1−θ̂)} (T̂)_{ad}) .

See the description given in [KS], p.49-50. Especially we can replace U by T_{sc} and S by T_{1} = T as explained in [KS], p. 49. (We are in the quasisplit case, and θ = θ^{∗}). The class V being already defined, let us explain the definition of A:

The element A is given by the endoscopic datum (^{L}M, M, s, ξ) of (G, θ). By transport of structure (twisted action) it is represented by hypercocycle with values in T̂_{G} → (T̂_{G})_{ad}. It is represented by a pair (a_{T}^{−1}, s) with the properties

1) s ∈ (T̂_{G})_{ad} is the image of the element s ∈ T̂_{G}, which comes from the endoscopic datum, under the map T̂_{G} → (T̂_{G})_{ad}.

2) a_{T} is the 1-cocycle in Z^{1}(W_{F} , T̂_{G}) such that ξ_{T_{M}} = a_{T} · ξ_{1}, where ξ_{T_{M}} is defined

by

ξ_{T_{M}} : ^{L}T_{M} → ^{L}M = M → ^{L}G ;

The first map is defined by the χ-data induced for M, the second by the embedding ξ determined by the endoscopic data. The second map ξ_{1} is defined by

ι ∘ ξ̌_{1} = ξ_{1} : ^{L}(T_{θ}) → ^{L}((Ĝθ̂)^{0}) = ^{L}G′ → ^{L}G .

The first arrow is again defined by (induced) χ-data. They only involve constructions in the group of fixed points (Ĝθ̂)^{0} ⊂ Ĝθ̂. In our case this group is connected and ^{L}((Ĝθ̂)) = ^{L}G′. The last map is the ’diagonal’ map, denoted ι.

This desciption applies to all ’three’ cases we considered. We denote the corresponding classes A^{E}, A^{F} and A respectively. Since all constructions involving the χ-data do not change, one obtains the only ’real’ dependency on E in the map ξ. Suppose ξ_{χ}^{M}(w) =

b(w) × w ∈ ^{L}M, b(w) ∈ M̂. Then

ξ_{E}(ξ_{χ}^{M}(w)) = ξ_{E}(b(w) × w) = ι(b(w))ξ_{E}(w) = a^{E}(w)ιξ̌_{1}(w) .

On the other hand b(w)ξ(w) = a(w)ξ̌_{1}(w), if we view M̂ as a subgroup of Ĝ. Therefore

ι(b(w))ιξ(w) = ι(a(w))ιξ̌_{1}(w) .

This implies

Int(ι(b(w)))(ιξ(w)(ξ^{E})^{−1}(w)) = ι(a(w))(a^{E}(w))^{−1} ∈ T̂_{G} .

Suppose ξ(w) = c(w) × w. From our assumptions s ∈ Z(M̂)^{Γ} and Int(s)ξ = ξ we get

sσ_{F}^{j} (c(w)) = σ_{F}^{j} (c(w))s for all j. Put c = diag(c(w), ..., c(w)). Then

ξ^{E}(σ_{F}) = s_{E}c × σ_{F} .

This implies

ξ^{E}(σ_{F}^{i}) = (s_{E}c)σ_{F} (s_{E}c)...σ_{F}^{i−1}(s_{E}c) × σ_{F}^{i}
= (s_{E})σ_{F} (s_{E})...σ_{F}^{i−1}(s_{E})cσ_{F} (c)...σ_{F}^{i−1}(c) × σ_{F}^{i} .

On the other hand

ιξ(w) = cσ_{F} (c)...σ_{F}^{i−1}(c) × σ_{F}^{i} .

Therefore

ιξ(w)(ξ^{E})^{−1}(w) = ((s_{E})σ_{F} (s_{E})...σ_{F}^{ord(w)−1}(s_{E}))^{−1} = (s_{E})θ(s_{E})...θ^{ord(w)−1}(s_{E})^{−1} .

where ord : W_{F} → (W_{F})^{ab} ≅ F^{∗} → ZZ is induced by the additive valuation map F^{∗} → ZZ. All these terms commute with ι(b(w)), because s ∈ Z(M̂). Therefore

a^{E}(w) = t^{s_{E}}ι(a(w)) ,

where t^{s_{E}}(w) is the unramified 1-cocycle

t^{s_{E}}(w) = s_{E}^{ord(w)} ∈ T̂_{G}^{Γ} ⊂ T̂_{G}

factoring over ord : W_{F} → W_{F}^{ab} ≅ F^{∗} → ZZ determined by t^{s_{E}}(σ_{F}) = s_{E} ∈ T̂_{G}. This gives the following elements a_{T}

a_{T}^{E}(w) = t^{s_{E}}(w) · ι(a_{T′}(w)) , case E

a_{T′}^{F}(w) = t^{s}(w) · a_{T′}(w) , case E = F

a_{T′}(w) , starting case .

The corresponding elements s in the hypercocycle are

s_{E} , case E ,

s , case E = F ,

s , starting case .

The hypercocycles are

(ι(a_{T′}(w))^{−1}, 1) · (t^{E}(w)^{−1}, s_{E}) = (ι(a_{T′}(w))^{−1}t^{E}(w)^{−1}, s_{E}) , case E ,

(a_{T′}(w)^{−1}, s) · (t^{F} (w)^{−1}, 1) = (a_{T′}(w)^{−1}t^{F} (w)^{−1}, s) , case E = F ,

(a_{T′}(w)^{−1}, s) = (a_{T′}(w)^{−1}, 1) · (1, s) , starting case ,

\underline{Pullbacks}: One has for K^{•} = (A → B) and K̂^{•} = B̂ → Â

0 → ((B̂)^{Γ})^{0} → H^{1}(W_{F} , K̂^{•}) → Hom_{cont}(H^{1}(F, K^{•})) → 0 .

These exact sequences are functorial with respect to ϕ : K_{1}^{•} → K_{2}^{•}. Suppose A ∈ H^{1}(W_{F} , K̂_{1}^{•}) is of the form A = ϕ̂_{∗}(X), then

⟨A, V⟩ = ⟨ϕ̂_{∗}(X), V⟩ = ⟨A, ϕ_{∗}(V)⟩ .

We may apply this to the morphism ϕ = (id, Nm_{E/F})

\begin{matrix}T_{sc} & →^{(1−θ)π} & T \\ ↓ id & & ↓ Nm_{E/F} \\ T_{sc} & →^{0} & T′ \end{matrix} .

Namely ι(a_{T′}(w)^{−1}, 1) = ˆ (id, Nm_{E/F})_{∗}(a_{T′}(w)^{−1}, 1), whereas

(id, Nm_{E/F})_{∗}(V) = [(v_{i}(σ), t)] .

Note H^{1}(A→^{0}B) = H^{1}(A) ⊕ H^{0}(B). Therefore

⟨ι[(a_{T′}(w)^{−1}, 1)], V⟩ = ⟨[(a_{T′}(w)^{−1}, 1)], [(v_{i}(σ), t)]⟩
= ⟨[(a_{T′}(w)^{−1}], t⟩⟨1, [v_{i}(σ)]⟩ = ⟨[(a_{T′}(w)^{−1}], t⟩
= Δ_{III_{2}}(γ_{M}, t) = Δ_{III_{2}}(γ_{M}, t)Δ_{III_{1}}(γ_{M}, t)
= Δ_{III}(γ_{M}, t) .

The factor Δ_{III_{1}}(γ_{M}, t) is trivial, because t is the image of γ_{M} under an admissable embedding, by assumption. We also used [KS], (A.3.13), which shows, that ⟨[(a^{−1}], t⟩ is the Langlands reciprocity pairing ⟨[a], t⟩_{L}.

Therefore assertion 2 is equivalent to

\underline{Assertion 2’:} Let the assumptions be those of assertion 2. Then

⟨[(t^{s_{E}})^{−1}, s_{E})], [(v_{i}(σ), δ^{∗})]⟩ = ⟨α(δ_{i}; t), s⟩ .

We simplify this statement. Note that

t^{s_{E}}(w) = (1, ..., 1, s) ∈ T̂_{G}
s_{E} = (1, ..., 1, s) ∈ (T̂_{G})_{ad} = T̂_{G}/Z(Ĝ) .

These modules are endowed with the twisted galois action, transported from T̂_{G}, (T̂_{G})_{ad} ≅ ˆ (T_{sc}). Note (T̂_{G})_{ad} = T̂_{G}/Z(Ĝ) ≅ ˆ ((T_{G})_{sc}) (see [B], p.44). Now use our assumption s ∈ Z(M̂)^{Γ}. This implies that s ∈ T̂_{M} = T̂_{G′} is invariant under conjugation by elements in M̂ and invariant under W_{F} (with the L-action on M̂). This implies s ∈ T_{M} is also invariant under the twisted action transported from T̂_{M}, because it only involves additional conjugations in M̂! This is the same as the twisted action on T̂_{G′} (transported from T′). Therefore s ∈ T̂′ is invariant. This implies, that the pair

t^{s_{E}}(w) = (1, ..., 1, s) ∈ T̂_{G}
s_{E} = (1, ..., 1, s) ∈ T̂_{G}

defines a hypercocycle in Z^{1}(W_{F} , T̂_{G} ⟶^{1−θ} T̂_{G}). Namely (1, ..., 1, s)^{1−θ} = (1, .., 1, s^{−1}, s),

whereas (1, ..., 1, s)σ_{F} (1, ..., 1, s)^{−1} = (1, ..., σ_{F} (s)^{−1}, s) = (1, ..., s^{−1}, s) in T̂_{G}. We use, that

σ_{F} (s) = s (twisted or untwisted action on T̂_{G′}), as explained above. The cocycle t^{s_{E}} is continuous by the definition of the topology on W_{F}.

We can consider the pullback with respect to the morphism ϕ = (π, id)

\begin{matrix}T_{sc} & →^{(1−θ)π} & T \\ ↓ π & & ↓ id \\ T & →^{1−θ} & T \end{matrix} .

This reformulates assertion 2’ into

\underline{Assertion 2”}: Let the assumptions be those of assertion 2. Then we have for the hypercohomology pairing between H^{1}(W_{F} , T̂ ⟶^{1−θ̂} T̂) and H^{1}(F, T ⟶^{1−θ} T)

⟨[(t^{s_{E}})^{−1}, s_{E})], inv(t, δ_{i})⟩ = ⟨α(t; δ_{i}), s⟩ .

\underline{Lemma 7}: If assertion 2,2’,2” holds for one δ_{i}, then for all δ = δ_{j} in its stable θ-conjugacy class.

\underline{Proof}: Both ⟨α(t; δ_{i}), s⟩ and ⟨[(t^{s_{E}})^{−1}, s_{E})], inv(t, δ_{i})⟩ have the same transformation property, if one replaces δ_{i} ∈ G(F) by δ ∈ G(F) within the same stable θ-conjugacy class. This is readily seen from going back to assertion 2. [KS], Thm 5.1.D gives Δ(M, G, t, δ) = Δ(M, G, t, δ_{i})⟨inv(δ_{i}, δ), s⟩. On the other hand we have already seen in the reduction of

lemma 1 to lemma 2, that α(t; δ) = α(t; δ_{i})⟨inv(δ_{i}, δ), s⟩.

Consider the

\underline{Special case}: Assume δ^{∗} with N_{θ}(δ^{∗}) = t can be chosen in T (F) = T′(F). We then also write δ^{∗} = δ.

\underline{Lemma 8}: Suppose res : H^{1}(F, T′) → H^{1}(F, T)^{θ} is surjective. Then we are in the special case.

\underline{Proof}: Under the assumption made it follows from the exact sequences in section 11, that every t ∈ H^{0}(F, T_{θ}) in the image of H^{1}(F, T → T) is also in the image of Norm_{E/F} :

T′(E) = T (F) → T′(F) ≅ T′(F).

In this special case one can reduce assertion 2” to the case δ_{i} = δ by lemma 7. So we will assume δ_{i} = δ = δ^{∗} in the special case, without any restriction of generality. With

δ_{i} = δ we get α(t; δ) = [c^{−1} · δ · σ_{F} (c)] = [δ] ∈ B(T′) = T′(L)/ ≃, because c = 1. Furthermore

we have in this special case inv(t; δ) = [(1, δ)] ∈ H^{1}(W_{F} , T →^{1−θ} T), because g_{i} = 1.

The equality of the right and left side in assertion 2” (special case) therefore boils down to ⟨[δ], s⟩ = ⟨t^{s_{E}}, δ⟩_{L}. Use [KS], (A.3.13). Note, these pairings are different pairings. On the right we have the Langlands pairing for the torus T/F, on the left the pairing between B(T′) and T̂^{′Γ}. The pairing on the left does depend on E only in the way that δ ∈ T′(E) ⊂ T′(L). We also may make a base extension and replace F by E, T′/F by T′/E and observe s ∈ T̂^{′Γ_{F}} ⊂ T̂^{′Γ_{E}}. In order to compare both sides, we can rewrite the right side as the Langlands pairing of the torus T′/E using Shapiro’s lemma.

\underline{Claim}: The Langlands pairing is compatible with Shapiro’s lemma.

\underline{Proof}: We may work with W_{K/F} for a normal splitting field extension K/F of T containing E. Let G = Γ_{K/F} be its Galois group. It is enough to check, that the maps in

0 →H_{T}^{−1}(G, Hom(\overline{K}^{∗}, T̂)) → Hom(\overline{K}∗, T̂)_{G} →^{N_{G}} Hom(\overline{K}∗, T̂)^{G} →H_{T}^{0} (G, Hom(\overline{K}^{∗}, X^{∗}(T̂)), C^{∗}))
↓ ∪u  ↓ cor  ↓ id  ↓ ∪u
0 →  H^{1}(G, T̂)  →^{Inf} H^{1}(W_{K/F} , T̂) →^{res} H^{1}(\overline{K}^{∗}, T̂)^{G} →  H^{2}(G, T̂)

(see [Milne] (8.6.1)) except the correstriction map cor respect restriction maps from W_{F} to W_{E} and therefore Shapiro’s isomorphism (see the description below). The vertical maps define isomorphisms (see [Milne], lemma 8.3). One is reduced to show, that the cup products with the fundamental class u respect these restrictions. This is well known. See [Milne], (0.1.5) and [Milne], Theorem 0.2.

The Shapiro isomorphism for induced moduls I = Ind(U, G, A)

H^{q}(G, I) ≅ H^{q}(U, A)

is given by the composite of the two maps

res : H^{q}(G, I) → H^{q}(U, I)
π : H^{q}(U, I) → H^{q}(U, A)

where π : I = ⊕_{g∈G/U}A is the projection on the g = 1 component. π is U equivariant. See Neukirch, p.68.

If we apply this to δ ∈ H^{0}(F, T (F)), we see that the image is δ ∈ H^{0}(E, T′(F)). If we apply this to t^{s_{E}} ∈ H^{1}(W_{F} , T̂), the image under restriction in H^{1}(W_{E}, T̂) is the cocycle t^{(s,s,..,s)}, determined by σ_{E} = (σ_{F})^{n} ↦ N_{σ_{F}} ((1, ..., 1, s)) = (s, s, ..., s). After projection under π we get the image t^{s} ∈ H^{1}(W_{E}, T̂′). Therefore ⟨t^{s_{E}}, δ⟩_{L,T/F} = ⟨t^{s}, δ⟩_{L,T′/E}. Assertion 2” in the special case is therefore equivalent to the assertion A(T′, E), described as follows

\underline{Assertion A(T, F)}: Consider the category of tori over a p-adic field F. There are natural transformations

ψ_{i} = T̂^{Γ} → H^{1}(W_{F} , T̂)

between the two functors T ↦ T̂^{Γ} and T ↦ H^{1}(W_{F} , T̂), defined by

ψ_{1} : T̂^{Γ} → Hom_{elm}(T (F), C^{∗}) ⊂ Hom_{cont}(T (F), C^{∗}) ≅ H^{1}(W_{F} , T̂)

ψ_{2} : T̂^{Γ} ≅ H^{1}(ZZ, T̂^{Γ}) →^{infl} H^{1}(W_{F} , T̂^{Γ}) → H^{1}(W_{F} , T̂) .

Inflation is with respect ord : W_{F} → ZZ. If ψ_{1} = ψ_{2}, we say that A(T, F) holds (Langlands reciprocity normalized properly).

Note that the two functors are related by

0 → T̂_{3}^{Γ} → T̂_{2}^{Γ} → T̂_{1}^{Γ} → H^{1}(W_{F} , T̂_{3}) → H^{1}(W_{F} , T̂_{2}) → H^{1}(W_{F} , T̂_{1}) → 0

for short exact sequences 0 → T_{1} → T_{2} → T_{3} → 0 of F-tori. Namely T̂^{Γ} = T̂^{W_{F}}, because W_{F} act via Γ on T̂. Surjectivity on the right follows from Langlands reciprocity because of extandability of continuous characters.

\underline{Lemma 9}: Suppose 0 → T_{1} → T_{2} → T_{3} → 0 with T_{2} ≅ ∏_{i} Res_{L_{i}/F} (G_{m}) and H^{1}(F, T_{1}) = 0.

Then A(T_{3}, F) holds.

\underline{Proof}: A(T_{2}, F) holds for T_{2} = G_{m}, a well known case. Then, by Shapiro’s lemma, it holds for T_{2}. From H^{1}(F, T) = 0 it follows, that H^{1}(W_{F} , T̂_{3}) → H^{1}(F, T̂_{2}) is injective. Therefore A(T_{3}, F) follows from A(T_{2}, F).

\underline{Remark}: The assumption of Lemma 9 is satisfied for all maximal tori T_{M} in M. Therefore lemma 9 proves A(T′, E) for all tori T′ ≅ T_{M} in M = Gl(2, F)^{2}/F^{∗}.

13.Final argument for G′ = GSp(4, F)

We now specialize the considerations of the last section to the case G′ = GSp(4, F) and the elliptic nontrivial endoscopic group considered so far.

Suppose δ ∈ G(F), t ∈ T′(F) and N_{θ}(δ^{∗}) = t for δ^{∗} ∈ T′(\overline{F}) with δ, δ^{∗} being θ-conjugate over \overline{F}.

In view of lemma 8 and lemma 9 and the remark after lemma 9 assertion 2,2’,2” is valid for the situation considered now, whenever the following holds:

(*) O_{δθ}^{G,κ}(f)) ≠ 0 implies t ∈ Norm_{E/F} (T′(E)).

If this holds, one can choose δ^{∗} ∈ T (F) and apply lemma 7, lemma 8, lemma 9.

By lemma 8 condition (*) is trivially satisfied for all tori T′ ≅ T_{M} except the tori T_{M} ≅ (L^{∗} × L^{∗})/F^{∗} ≅ L^{∗} × N_{L}^{1} or T_{M} ≅ (L_{1}^{∗} × L_{2}^{∗})/F^{∗}. For the first kind of tori, there are two cases

1) E is linear disjoint from L/F. Then (*) is automatically satisfied if n = [E : F] is odd.

2) E containes either L. Then (*) is automatically satisfied.

For the second kind of tori, there are three cases

1) E is linear disjoint from L_{1}L_{2}/F. Then (*) is automatically satisfied if n = [E : F] is odd.

2) E containes either L_{1} or L_{2}. Then (*) is automatically satisfied.

3) E containes E^{+}, but neither L_{1} or L_{2}. Then T/E is isomorphic to a torus ((L_{1}E)^{∗} × (L_{1}E)^{∗})/E^{∗} of first type over E. Condition (*) is satisfied if n = [E : F] is odd.

For the last case 3) observe

T̂_{M}^{Γ_{E}} = T̂_{M}^{Γ_{E/E^{+}}} = {(t_{1}, t_{2}, t_{3}) | σ(t_{1}, t_{2}, t_{3}) = (t_{2}t_{3}, t_{1}t_{3}, t_{3}^{−1}) = (t_{1}, t_{2}, t_{3})} .

This implies T̂_{M}^{Γ_{E}} = Z(Ĝ′) ∪ (1, −1, −1)Z(Ĝ′). Therefore T̂_{M}^{Γ_{E}} = T̂_{M}^{Γ_{F}}.

\underline{Consequence}: If (*) is not automatically satisfied, then L/F resp. L_{1}/F, L_{2}/F are ramified and [E : F] is even. Remember E/F is unramified.

So let us discuss the condition t ∈ Norm_{E/F} (T′(E)) of (*) in these remaining cases. We have for L_{i}/F ramified over F, E/F unramified of degree n = [E : F] divisible by two:

1  ↦  (2, −2)
ZZ/nZZ  →  ZZ/nZZ × ZZ/nZZ  → ZZ/nZZ × ZZ/2ZZ
↑ ord_{F}  ↑ ord_{L_{1}} × ord_{L_{2}}  ↑
1 →  F^{∗}  →  L_{1}^{∗} × L_{2}^{∗}  →  T (F)  → 1
↑ N  ↑ N  ↑ N
1 →  E^{∗}  →  (EL_{1})^{∗} × (EL_{2})^{∗}  →  T (E)  → 1

by Hilbert 90. The vertical maps are the norm maps Norm_{E/F}. Because EL_{i}/L_{i} and E/F are unramified field extensions, the image of the norm map is the group of elements of order divisible by n. Therefore in order to verify (*) one has to show

(**) O_{δθ}^{G,κ}(f)) ≠ 0 with t_{M} = (t_{1}, t_{2}) mod (t, t^{−1}), t ∈ F^{∗} implies, that there exists a

t ∈ F^{∗} such that t_{i} ∈ L_{i}^{∗} satisfy ord_{L_{1}}(t_{1}t) ∈ [E : F] · ZZ and ord_{L_{2}}(t_{2}t^{−1}) ∈ [E : F] · ZZ

(normalized such that ord_{L_{i}}(π_{L_{i}}) = 1 and therefore ord(π_{F}) = 2).

The proof of the twisted findamental lemma and lemma 2 have been reduced to the proof of (**). One has only to verify (**) in the nontrivial ’ramified’ cases listed above. In order to do this we use a bootstrap argument. We first consider (**) in the case f = 1_{K} and local fields F of residue characteristic different from two. We will show below, that condition (**) follows in these cases from lemma 5. As already explained this proves lemma 1; and therefore by global methods the twisted fundamental lemma for all spherical function follows along the lines of Clozel and Labesse. It remains to establish lemma 2, which is necessary for the applications in [Ko,Shimura]. For that we apply the fundamental lemma to the spherical function f = ϕ_{n}. This shows, that

O_{δθ}^{G,κ}(f)) ≠ 0 ⇔ O_{t_{M}}^{M} (ϕ_{n}^{M})) ≠ 0 .

The endoscopic function ϕ_{n}^{M} attached to ϕ_{n} was already determined in section 3. To express it explicitely we saw, that it is most conveniant to pass to the z-extension M_{1} = Gl(2, F)^{2} of M. On M_{1} it is equal (up to some adjustments by (t · id, t^{−1} · id) to the function

f^{(n)}(t_{1}, t′_{1})1_{K_{Gl(2)}}(t_{2}, t′_{2}) − 1_{K_{Gl(2)}}(t_{1}, t′_{1})f^{(n)}(t_{2}, t′_{2}) .

The function f^{(n)} is a spherical function on Gl(2, F). It is the same as the function used in Casselman [Corv], lemma 5.3.1. Therefore the validity of (**) follows from the \underline{Lemma 10}: The orbital integrals O_{γ}^{Gl(2,F)}(f^{(n)}) for elliptic elements γ ∈ Gl(2, F) with eigenvalues t, t′ in a ramified quadratic extension field of F are nonzero iff ord_{L}(t) =

ord_{L}(t′) = n.

\underline{Proof}: This follows from prop.5.3.3 loc. cit. for n ≥ 1. The case n = 1 being trivial.

It remains to check the condition (**) for the initial step of the bootstrap, i.e. in the case of the unit element f = 1_{K} (for residue characteristic different from two):

\underline{1.Case}: (tori T = (L^{∗} × L^{∗})/F^{∗})

According to [W1,W3] the element t is determined by t_{M} = (t_{1}, t_{2}) via x = (t_{1}t_{2})^{σ} and x′ = t_{1}(t_{2})^{σ}. The eigenvalues of t are those of x and x′. If the orbital integral O_{t}^{G,κ}(1′_{K}) for the unit element 1_{K′} is nonzero (which follows from [Ko] and the assumption of (**)), the eigenvalues of t must be units. This implies ord_{L}(t_{1})+ord_{L}(t_{2}) = 0. It remains only to show, that ord(t_{2}) is even. Both together imply (**). But the condition ord(t_{2}) even is equivalent to the condition, that κ([t]) ∈ {±1} is equal to one. This follows from lemma 5.

\underline{2.Case}: (tori T = (L_{1}^{∗} × L_{2}^{∗})/F^{∗})

According to [W2] the element t is determined by t_{M} = (t_{1}, t_{2}) via x = t_{1}t_{2}. The eigenvalues of t are those of x. If the orbital integral O_{t}^{G,κ}(1′_{K}) for the unit element is nonzero (which follows again from [Ko] and the assumption of (**)), the eigenvalues of t must be units. This implies ord_{L}(t_{1}) + ord_{L}(t_{2}) = 0. It remains only to show, that ord(t_{2}) is even. Both together imply (**). But the condition ord(t_{2}) even is again equivalent to the condition, that κ([t]) ∈ {±1} is equal to one. See the section, preliminary considerations on x in [W1]. Again (**) therefore follows from lemma 5.