Remarks on the Fundamental Lemma for stable twisted Endoscopy of Classical Groups

Joachim Ballmann^{1} Rainer Weissauer

Uwe Weselmann^{1}

August 13, 2002

1supported by Deutsche Forschungsgemeinschaft

Contents

Introduction i 1 Stable endoscopy and matching 1 2 Centralizers 8 3 Topological Jordan decomposition 14 4 Classification of Θ-conjugacy classes 16 5 Orbital integrals 23 6 Comparison between PGL_{2n+1} and Sp_{2n} 28 7 Comparison between GL_{2n} × GL_{1} and GSpin_{2n+1} 35 8 Comparison between SO_{2n+2} and Sp_{2n} 43 Bibliography 50

Introduction

The original aim of these notes is to prove a fundamental lemma for the stable lift from H = Sp_{4} to G̃ = { \atop PGL}_{5} over a local non archimedean field F with residue characteristic ≠ 2. Here { \atop PGL}_{5} = PGL_{5} ⋊ ⟨Θ⟩ is generated by its normal subgroup PGL_{5} of index 2 and the involution Θ : g ↦ J ·^{t}g^{−1}·J^{−1}, where J is the antidiagonal matrix with entries (1, −1, 1, −1, 1).

We will (Cor.7.10) prove that if the semisimple elements γΘ ∈ { \atop PGL}_{5}(F) and η ∈ Sp_{4}(F) match (see 1.11 for a definition of matching) then the corresponding stable orbital integrals (see 5.1) for the unit elements in the Hecke algebra match:

O_{γΘ}^{st}(1, { \atop PGL}_{5}) = O_{η}^{st}(1, Sp_{4}). (i)

This theorem will have important applications in the theory of automorphic representations of the group GSp_{4} over a number field and for the l–adic Galoisrepresentations on the corresponding Shimura varieties [W1], [W2], [Wes].

In analyzing (i) using the Kazhdan-trick (lemma 5.5 below) we recognized that all essential computations had already been done by Flicker in [Fl2], where the corresponding fundamental lemma for the lift from GSp_{4} ≃ GSpin_{5} to G̃L_{4} × 𝔾_{m} has been proved. This phenomenon seems to be known to the experts [Hal3].

More generally one can discuss the fundamental lemma for the stable lift from H to a classical split group with outer automorphism G̃, where H is the stable endoscopic group for G̃. This fundamental lemma describes a relationship between ordinary stable orbital integrals on the endoscopic group H and Θ-twisted orbital integrals on G̃. We will discuss the following lifts in detail:

H  G̃
Sp_{2n}  PGL_{2n+1} ⋊ ⟨g ↦ J^{t}g^{−1}J^{−1}⟩
GSpin_{2n+1} (GL_{2n} × 𝔾_{m}) ⋊ ⟨(g, a) ↦ (J^{t}g^{−1}J^{−1}, det g · a)⟩
Sp_{2n}  { \atop SO_{2n+2}} ≃ O_{2n+2}.

In each case we will reduce the fundamental lemma using the Kazhdan trick and a lot of observations in linear algebra to a statement which we call the BC-conjecture and which seems to be proven only for n = 1, 2: Conjecture:If the regular topologically unipotent and algebraically semisimple elements u ∈ SO_{2n+1}(O_{F}) and v ∈ Sp_{2n}(O_{F}) are BC-matching (see 1.12) then

O_{u}^{st}(1, SO_{2n+1}) = O_{v}^{st}(1, Sp_{2n}). (BC_{n})

Thus the (Θ-twisted) fundamental lemmas for the three series of endoscopy will be reduced to a fundamental lemma like statement for ordinary (i.e. untwisted) stable orbital integrals on the groups SO_{2n+1} of type B_{n} resp. Sp_{2n} of type C_{n}. An outline of the proofs will be given in chapter 2.

1 Stable endoscopy and matching

(1.1) Notations. In this paper we will denote by F a p-adic field with ring of integers O_{F}, prime ideal 𝔭 and uniformizing element ϖ = ϖ_{F}. The residue field of characteristic p is denoted κ = κ_{F} = O_{F} /𝔭. By F̄ we denote an algebraic closure of F. In the whole paper we will assume that p ≠ 2. Only in this chapter R denotes an arbitrary integral domain.

(1.2) Split Groups with automorphism. Let G/R be a connected reductive split group scheme. We fix some ”splitting” i.e. a tripel (B, T, {X_{α}}_{α∈Δ}) where T denotes a maximal split-torus inside a rational Borel B, Δ = Δ_{G} = Δ(G, B, T) ⊂ Φ(G, T) ⊂ X^{∗}(T) the set of simple roots inside the system of roots and the X_{α} are a system (nailing) of isomorphisms between the additive group scheme 𝔾_{a} and the unipotent root subgroups B_{α}. Here X^{∗}(T) = Hom(T, 𝔾_{m}) denotes the character module of T, while X_{∗}(T) = Hom(𝔾_{m}, T) will denote the cocharacter module of T. Let Θ ∈ Aut(G) be an automorphism of G which fixes the splitting, i.e. stabilizes B and T and permutes the X_{α}. We assume Θ to be of finite order l. We denote by

G̃ = G ⋊ ⟨Θ⟩

the (nonconnected) semidirect product of G with Θ. Θ acts on the (co)character module via X_{∗}(T) ∋ α^{∨} ↦ Θ ∘ α^{∨} resp. X^{∗}(T) ∋ α ↦ α ∘ Θ^{−1}.

(1.3) The dual group. Let Ĝ = Ĝ(ℂ) be the dual group of G. By definition Ĝ has a tripel (B̂, T̂, {X̂α̂}) such that we have identifications X^{∗}(T̂) = X_{∗}(T), X_{∗}(T̂) = X^{∗}(T) which identifies the (simple) roots α̂ ∈ X^{∗}(T̂) with the (simple) coroots a^{∨} ∈ X_{∗}(T), and the (simple) coroots α̂^{∨} ∈ X_{∗}(T̂) with the (simple) roots α ∈ X^{∗}(T). There exists a unique automorphism Θ̂ of Ĝ which stabilizes (B̂, T̂, {X̂α̂}) and induces on (X_{∗}(T̂), X^{∗}(T̂)) the same automorphism as Θ on (X^{∗}(T), X_{∗}(T)).

(1.4) The Θ-invariant subgroup in Ĝ. Let Ĥ = (ĜΘ̂)^{∘} be the connected component of the subgroup of Θ̂-fixed elements in Ĝ. It is a reductive split group with triple (B̂_{H}, T̂_{H}, {X_{β}}_{β∈ΔĤ}), where B̂_{H} = B̂Θ̂, T̂_{H} = T̂ Θ̂ and the nailing will be explained soon. We have the inclusion of cocharacter modules X_{∗}(T̂_{H}) = X_{∗}(T̂)Θ̂ ⊂ X_{∗}(T̂) and a projection for the character module

P_{Θ} : X^{∗}(T̂) ↠ (X^{∗}(T̂)Θ̂)_{free} = X^{∗}(T̂_{H}),

where (X^{∗}(T̂)Θ̂)_{free} denotes the maximal free quotient of the coinvariant module X^{∗}(T̂)Θ̂. For a ℤ[Θ]-module X we define a map

S_{Θ} : X → X^{Θ}, x ↦ ∑_{i=0}^{ord_{x}(Θ)−1} Θ^{i}(x)

where ord_{x}(Θ) = min{i > 0 | Θ^{i}(x) = 0} is length of the orbit ⟨Θ⟩(x), which may vary on X. For the roots Φ and coroots Φ^{∨} of a given root datum (X^{∗}, X_{∗}, Φ, Φ^{∨}) we have to introduce a modified map S′_{Θ} by

S′_{Θ}(α) = c(α) · S_{Θ}(α)  where
c(α) = \frac{2}{⟨α^{∨}, S_{Θ}(α)⟩}

resp. by the formula where the roles of α and α^{∨} are exchanged. For all simple root systems with automorphisms which are not of type A_{2n} we have ⟨α^{∨}, Θ^{i}(α)⟩ = 0 for i = 1, …, ord_{α}(Θ) − 1 which implies c(α) = 1 i.e. S′_{Θ}(α) = S_{Θ}(α). We furthermore introduce the subset of short-middle roots and the dual concept of long-middle coroots:

Φ(Ĝ, T̂)^{sm} = \{α ∈ Φ(Ĝ, T̂) |\frac{1}{2} · P_{Θ}(α) ∉ P_{Θ}(Φ(Ĝ, T̂))\}
Φ^{∨}(Ĝ, T̂)^{lm} = {α^{∨} |α ∈ Φ(Ĝ, T̂)^{sm}}

Proposition 1.5. With the above notations we have

(i)  Φ(Ĥ, T̂_{H}) = P_{Θ}(Φ(Ĝ, T̂)^{sm})  for the roots
(ii)  Φ^{∨}(Ĥ, T̂_{H}) = S′_{Θ}(Φ^{∨}(Ĝ, T̂)^{lm})  for the coroots
Δ∨ ; Ĥ = Δ^{∨}(Ĥ, B̂_{H}, T̂_{H}) = S′_{Θ}(Δ∨ ; Ĝ)  for the simple coroots
ΔĤ = Δ(Ĥ, B̂_{H}, T̂_{H}) = P_{Θ}(ΔĜ)  for the simple roots

Proof: This follows from [St, 8.1], which is restated in Theorem 2.1 below.

Definition 1.6 (stable Θ-endoscopic group). In the above situation a connected reductive split group scheme H/R will be called a stable Θ-endoscopic group for (G, Θ) resp. G̃ if its dual group is together with the splitting isomorphic to the above (Ĥ, B̂_{H}, T̂_{H}, {X_{β}}_{β∈ΔĤ}).

Remarks: Since H is unique up to isomorphism (up to unique isomorphism if we consider H together with a splitting) we can call H the stable Θ-endoscopic group for (G, Θ). For a maximal split-torus T_{H} ⊂ H we have:

(iii)  X_{∗}(T_{H}) = X_{∗}(T)_{Θ}  for the cocharacter-module
X^{∗}(T_{H}) = X^{∗}(T)^{Θ}  for the character-module

(1.7) To get examples we use the following notations:

diag(a_{1}, …, a_{n}) ∈ GL_{n} denotes the diagonal matrix (δ_{i,j} · a_{i})_{ij} and

antidiag(a_{1}, …, a_{n}) ∈ GL_{n} the antidiagonal matrix (δ_{i,n+1−j} · a_{i})_{ij} with a_{1} in the upper right corner. We introduce the following matrix

J = J_{n} = (δ_{i,n+1−j}(−1)^{i−1})_{1≤i,j≤n} = antidiag(1, −1, …, (−1)^{n−1}) ∈ GL_{n}(R).

and its modification J′_{2n} = antidiag(1, −1, 1, …, (−1)^{n−1}, (−1)^{n−1}, …, 1, −1, 1). Since ^{t}J_{n} = (−1)^{n−1} · J_{n} and J′_{2n} is symmetric we can define the

standard symplectic group  Sp_{2n} = Sp(J_{2n})
standard split odd orthogonal group SO_{2n+1} = SO(J_{2n+1}).
standard split even orthogonal group  SO_{2n} = SO(J′_{2n}).

We consider the groups GL_{n}, SL_{n}, PGL_{n}, Sp_{2n}, SO_{n} with the splittings consisting of the diagonal torus, the Borel consisting of upper triangular matrices and the standard nailing. We remark that the following map defines an involution of GL_{n}, SL_{n} and PGL_{n}:

Θ = Θ_{n} : g ↦ J_{n} ·^{t}g^{−1} · J_{n}^{−1}.

Example 1.8 (A_{2n} ↔ C_{n}).

G = PGL_{2n+1}, Θ = Θ_{2n+1}  has dual  Ĝ = SL_{2n+1}(ℂ), Θ̂ = Θ_{2n+1} ; ⋃ ; H = Sp_{2n}  has dual  Ĥ = SO_{2n+1}(ℂ)

Example 1.9 (A_{2n−1} ↔ B_{n}). The group G = GL_{2n} × 𝔾_{m} has the automorphism

Θ : (g, a) ↦ (Θ_{2n}(g), det(g) · a)

which is an involution since det(Θ_{2n}(g)) = det g^{−1}. The dual Θ̂ ∈ Aut(Ĝ) satisfies

Θ̂(g, b) = (Θ_{2n}(g) · b, b),  so that we get

G = GL_{2n} × 𝔾_{m}, Θ  has dual  Ĝ = GL_{2n}(ℂ) × ℂ^{×}, Θ̂ ; ⋃ ; H = GSpin_{2n+1}  has dual  Ĥ = GSp_{2n}(ℂ).

Recall that GSpin_{2n+1} can be realized as the quotient (𝔾_{m} × Spin_{2n+1}) /μ_{2}, where μ_{2} ≃ {±1} is embedded diagonally, so that we get an exact sequence

1 → Spin_{2n+1} → GSpin_{2n+1} ⟶^{μ} 𝔾_{m} → 1,

where the ”multiplier” map μ is induced by the projection to the 𝔾_{m} factor followed by squaring. Thus the derived group of GSpin_{2n+1} is Spin_{2n+1}, i.e. a connected, split and simply connected group.

Example 1.10 (D_{n+1} ↔ C_{n}). We furthermore have the situation:

G = SO_{2n+2}, Θ = int(s)  has dual  Ĝ = SO_{2n+2}(ℂ), Θ̂ = int(ŝ) ; ⋃ ; H = Sp_{2n}  has dual  Ĥ = SO_{2n+1}(ℂ)

where s ∈ O_{2n+2} denotes the reflection which interchanges the standard basis vectors e_{n+1} and e_{n+2} and fixes all other basis elements e_{i}. The ŝ is of the same shape.

(1.11) Matching elements Since each semisimple Θ-conjugacy resp. conjugacy class in G(F̄) resp. H(F̄) meets T (F̄) resp. T_{H}(F̄) we have canonical bijections

i_{G} : G(F̄)_{ss}/Θ − conj  ≃ T (F̄)_{Θ}/(W_{G})^{Θ}
i_{H} : H(F̄)_{ss}/conj  ≃ T_{H}(F̄)/W_{H}

where W_{G} = Norm(T, G)/T and W_{H} = Norm(T_{H}, H)/T_{H} denote the Weyl-groups. We further have an isomorphism

N_{KS} : T (F̄)_{Θ} ≃ (X_{∗}(T) ⊗ F̄^{×})_{Θ} ≃ X_{∗}(T)_{Θ} ⊗ F̄^{×} = X_{∗}(T_{H}) ⊗ F̄^{×} ≃ T_{H}(F̄)

and observe W_{H} ≃ (W_{G})^{Θ}. Therefore we may define: Two (Θ-)semisimple elements γΘ ∈ G(F)Θ and h ∈ H(F) are called matching if their (Θ-)stable conjugacy classes in G(F̄) resp. H(F̄) correspond to each other via the isomorphism i_{H}^{−1} ∘ N_{KS} ∘ i_{G}.

(1.12) BC-matching: We have an isomorphism between the standard diagonal tori:

i_{BC} : T_{SO_{2n+1}} → T_{Sp_{2n}},
diag(t_{1}, …, t_{n}, 1, t_{n}^{−1}, …, t_{1}^{−1}) ↦ diag(t_{1}, …, t_{n}, t_{n}^{−1}, …, t_{1}^{−1}).

We observe that i_{BC} induces an isomorphism of Weylgroups:

W_{SO_{2n+1}} ≃ S_{n} ⋉ {±1}^{n} ≃ W_{Sp_{2n}}

Two semisimple elements h ∈ SO_{2n+1}(F) and b ∈ Sp_{2n}(F) are called BC-matching if their stable conjugacy classes correspond to each other under the isomorphism i_{Sp_{2n}}^{−1} ∘ i_{BC} ∘ i_{SO_{2n+1}}.

Example 1.13. In example 1.8 above the norm map N_{KS} : T → T_{Θ} ≃ T_{H} is given by

(iv)  γ = diag(t_{1}, t_{2}, …, t_{n}, t_{n+1}, t_{n+2}, …, t_{2n+1})  ∈ T ⊂ PGL_{2n+1}
↦ h = diag(t_{1}/t_{2n+1}, t_{2}/t_{2n}, …, t_{n}/t_{n+2}, t_{n+2}/t_{n}, …, t_{2n+1}/t_{1}) ∈ T_{H} ⊂ Sp_{2n}.

Proof: We identify X_{∗}(T) ≃ ℤ^{2n+1}/ℤ ≃ ⨁_{i=1}^{2n+1} ℤf_{i}/ℤ ∑_{i=1}^{2n+1} f_{i} and X^{∗}(T) ≃ {∑_{i=1}^{2n+1} α_{i}e_{i} | ∑_{i=1}^{2n+1} α_{i} = 0}, such that

f_{i} : t ↦ diag(1, …, 1, t_{i}, 1, …, 1) ∈ T  and
e_{i} − e_{j} : T ∋ diag(t_{1}, …, t_{2n+1}) ↦ t_{i}/t_{j}.

Similarly we identify T_{H} ≃ 𝔾_{m}^{n} via diag(t_{1}, …, t_{n}, t_{n}^{−1}, …, t_{1}^{−1}) ↦ (t_{i})_{1≤i≤n} and write X_{∗}(T_{H}) ≃ X_{∗}(𝔾_{m}^{n}) ≃ ⨁_{i=1}^{n} ℤf′_{i} resp. X^{∗}(T_{H}) ≃ X^{∗}(T_{H}) ≃ ⨁_{i=1}^{n} ℤe′_{i}. The involution Θ acts as

Θ(f_{i}) = −f_{2n+2−i}, Θ(e_{i} − e_{j}) = e_{2n+2−j} − e_{2n+2−i}

Now it is clear that we have an identification P_{Θ} : X_{∗}(T)_{Θ} ≃ X_{∗}(T_{H}) given by P_{Θ}(f_{i}) = f′_{i} and P_{Θ}(f_{2n+2−i}) = −f′_{i} for 1 ≤ i ≤ n. This forces P_{Θ}(f_{n+1}) = P_{Θ}(− ∑_{i≠n+1} f_{i}) = P_{Θ}(Θ(∑_{i=1}^{n} f_{i})−∑_{i=1}^{n} f_{i}) = 0. Dual to this we have an injection ι : X^{∗}(T_{H}) ≃ X^{∗}(T)^{Θ} ⊂ X^{∗}(T) such that ι(e′_{i}) = e_{i} −e_{2n+2−i}. It is furthermore clear that this P_{Θ} induces the map (iv) The claim now follows if show that P_{Θ} and ι correspond to the natural maps on the side of the dual groups which are characterized by the equations of Proposition 1.5. Especially we have to check the duals of the relations (i) and (ii) for our explicitly given maps P_{Θ} and ι, namely the equations

(v)  Φ^{∨}(H, T_{H}) = P_{Θ}(Φ^{∨}(G, T)^{sm})  for the coroots
(vi)  ι(Φ(H, T_{H})) = S′_{Θ}(Φ(G, T)^{lm})  for the roots.

But we have

(vii)  P_{Θ}(±(f_{i} − f_{j})) = ±(f′_{i} − f′_{j}) = P_{Θ}(±(f_{2n+2−j} − f_{2n+2−i})),
(viii) P_{Θ}(±(f_{i} − f_{2n+2−j})) = ±(f′_{i} + f′_{j}),
P_{Θ}(±(f_{i} − f_{n+1})) = ±f′_{i} = P_{Θ}(±(f_{n+1} − f_{2n+2−i}))

where 1 ≤ i, j ≤ n in all three equations, but where additionally i ≠ j in (vii) while i = j is allowed in (viii). Nevertheless P_{Θ}(±(f_{i} − f_{2n+2−i})) = 2 · f′_{i} is not a member of the right hand side of (v) since f′_{i} = P_{Θ}(f_{i} −f_{n+1}). By the well known description of Φ^{∨}(Sp_{2n}, T_{H}) we get the equality (v). Similarly we get

S_{Θ}(±(e_{i} − e_{j})) = ι(±(e′_{i} − e′_{j})) = S_{Θ}(±(e_{2n+2−j} − e_{2n+2−i})),
S_{Θ}(±(e_{i} − e_{2n+2−j})) = ι(±(e′_{i} + e′_{j})),
S_{Θ}(±(e_{i} − e_{n+1})) = ±(e_{i} − e_{2n+2−i}) = S_{Θ}(±(e_{n+1} − e_{2n+2−i}))
= ±ι(e′_{i}) = S_{Θ}(±(e_{i} − e_{2n+2−i}))

where 1 ≤ i ≠ j ≤ n in the first two equations and 1 ≤ i ≤ n in the last two. Since Φ(Sp_{2n}, T_{H}) = {e′_{i} − e′_{j} | 1 ≤ i ≠ j ≤ n} ∪ {e′_{i} + e′_{j} | 1 ≤ i ≠ j ≤ n} ∪ {2 · e′_{i} | 1 ≤ i ≤ n} and since ⟨f_{i} − f_{n+1}, S_{Θ}(e_{i} − e_{n+1})⟩ = 1 but ⟨f_{i} − f_{j}, S_{Θ}(e_{i} − e_{j})⟩ = 2 for all i, j ≠ n + 1, 1 ≤ i, j ≤ 2n + 1 we get the claim (vi) in view of the definition of S′_{Θ}. Finally it is clear, that S′_{Θ} maps the set of simple roots {e_{i} − e_{i+1}, e_{n} − e_{n+1}, e_{n+1} − e_{n+2}, e_{2n+1−i} − e_{2n+2−i} | 1 ≤ i ≤ n − 1} of G to the set of simple roots ι({e′_{i} − e′_{i+1}, 2 · e′_{n} | 1 ≤ i ≤ n − 1}) of H and that P_{Θ} maps the set of simple coroots {f_{i} − f_{i+1}, f_{n} − f_{n+1}, f_{n+1} − f_{n+2}, f_{2n+1−i} − f_{2n+2−i} | 1 ≤ i ≤ n − 1} of G to the set of simple coroots {f′_{i} − f′_{i+1}, f′_{n} | 1 ≤ i ≤ n − 1}.

Example 1.14. In example 1.9 above we consider additionally the projection pr_{ad} : GSpin_{2n+1} → SO_{2n+1} = Spin_{2n+1}/{±1}. Then the composite map pr_{ad} ∘ N_{KS} : T → T_{ad} ⊂ SO_{2n+1} is given by

(ix)  γ = (diag(t_{1}, t_{2}, …, t_{n}, t_{n+1}, …, t_{2n}), t_{0})  ∈ T ⊂ GL_{2n} × 𝔾_{m}
↦ h = diag(t_{1}/t_{2n}, …, t_{n}/t_{n+1}, 1, t_{n+1}/t_{n}, …, t_{2n}/t_{1}) ∈ T_{ad} ⊂ SO_{2n+1}.

Proof: We consider the following basis (e_{i})_{0≤i≤2n} of X^{∗}(T):

e_{i} : T ∋ (diag(t_{1}, …, t_{2n}), t_{0}) ↦ t_{i}.

Let (f_{i})_{0≤i≤2n} be the dual basis of X_{∗}(T). We furthermore identify T_{ad} ≃ 𝔾_{m}^{n} via diag(t_{1}, …, t_{n}, 1, t_{n}^{−1}, …, t_{1}^{−1}) ↦ (t_{i})_{1≤i≤n} and write X_{∗}(T_{ad}) ≃ X_{∗}(𝔾_{m}^{n}) ≃ ⨁_{i=1}^{n} ℤf̃_{i} resp. X^{∗}(T_{H}) ≃ X^{∗}(T_{H}) ≃ ⨁_{i=1}^{n} ℤẽ_{i}. The involution Θ acts via

Θ(e_{i}) = −e_{2n+1−i},  Θ(f_{i}) = −f_{2n+1−i} + f_{0} for 1 ≤ i ≤ 2n
Θ(e_{0}) = e_{0} + ∑_{i=1}^{2n} e_{i},  Θ(f_{0}) = f_{0}

Now it is clear that X^{∗}(T_{H}) = X^{∗}(T)^{Θ} has as basis (e′_{0}, e′_{1}, …, e′_{n}) where e′_{i} = e_{i} − e_{2n+1−i} for 1 ≤ i ≤ n and e′_{0} = e_{0} + ∑_{i=n+1}^{2n} e_{i}. Let (f′_{0}, f′_{1}, …, f′_{n}) be the dual basis of X_{∗}(T_{H}) = X_{∗}(T)_{Θ}. Then the projection map P_{Θ} : X_{∗}(T) → X_{∗}(T)_{Θ} satisfies: P_{Θ}(f_{0}) = f′_{0}, P_{Θ}(f_{i}) = f′_{i}, P_{Θ}(f_{2n+1−i}) = −f′_{i} + f′_{0} for 1 ≤ i ≤ n. For the (co)root systems we get:

Φ(GSpin_{2n+1}, T_{H}) = P_{Θ}(Φ(GL_{2n} × 𝔾_{m}, T))
= {±e′_{i} ± e′_{j} | 1 ≤ i < j ≤ n} ∪ {e′_{i} | 1 ≤ i ≤ n}
Φ^{∨}(GSpin_{2n+1}, T_{H}) = S′_{Θ}(Φ^{∨}(GL_{2n} × 𝔾_{m}, T)),
= {±(f′_{i} − f′_{j}) | 1 ≤ i < j ≤ n} ∪ {±(f′_{i} + f′_{j} − f′_{0}) | 1 ≤ i ≤ j ≤ n}.

The cocharacter group of the center of GSpin_{2n+1} is then recognized as ℤf′_{0}. Thus we may define a surjection pr_{ad} : X_{∗}(T_{H}) → X_{∗}(T_{ad}) by f′_{0} ↦ 0 and f′_{i} ↦ f̃_{i} for 1 ≤ i ≤ n. Dually one has the injection ι : X^{∗}(T_{ad}) ↪ X^{∗}(T_{H}), ẽ_{i} ↦ e′_{i}. Now it is clear that pr_{ad} ∘ P_{Θ} induces the map (ix) and it remains to show that we have the following relations analogous to (v) and (vi):

(x)  Φ^{∨}(SO_{2n+1}, T_{ad}) = pr_{ad} ∘ P_{Θ}(Φ^{∨}(G, T)^{sm})  for the coroots
(xi) ι(Φ(SO_{2n+1}, T_{ad})) = S′_{Θ}(Φ(G, T)^{lm})  for the roots.

But this follows immediately from the above description of P_{Θ}(Φ(GL_{2n} × 𝔾_{m}, T)) and S′_{Θ}(Φ^{∨}(GL_{2n} × 𝔾_{m}, T)) in view of the very simple shape of pr_{ad} and ι and the knowledge of the (co)root system of SO_{2n+1}. The relation for the bases of the (co)root systems is checked in a similar way.

Example 1.15. In example 1.9 above we now analyze the relation between the multiplier map μ and matching. We claim: If (h, a) ∈ GL_{2n}(F) × F^{×} and η ∈ GSpin_{2n+1}(F) match then we have:

μ(η) = det(h) · a^{2}.

Proof: In the notations of 1.14 the element e′ = 2e′_{0} + ∑_{i=1}^{n} e′_{i} = 2e_{0} + ∑_{i=1}^{2n} e_{i} ∈ X^{∗}(T_{H}) = X^{∗}(T)^{Θ} corresponds to the character (h, a) ↦ det(h) · a^{2}. Since e′ is orthogonal to the coroots Φ^{∨}(GSpin_{2n+1}, T_{H}) it has to correspond to a multiple of the multiplier μ. Now it is easy to see that it corresponds in fact to μ.

Example 1.16. In example 1.10 above the norm map N_{KS} : T → T_{Θ} ≃ T_{H} is given by

(xii)  γ = diag(t_{1}, t_{2}, …, t_{n}, t_{n+1}, t_{n+1}^{−1}, …, t_{1}^{−1})  ∈ T ⊂ SO_{2n+2}
↦ h = diag(t_{1}, …, t_{n}, t_{n}^{−1}, …, t_{1}^{−1}) ∈ T_{H} ⊂ Sp_{2n}.

Proof: We consider the following basis (e_{i})_{1≤i≤n+1} of X^{∗}(T):

e_{i} : T ∋diag(t_{1}, …, t_{n+1}, t_{n+1}^{−1}, …, t_{1}^{−1}) ↦ t_{i}.

Let (f_{i})_{1≤i≤n+1} be the dual basis of X_{∗}(T). The involution Θ acts via

Θ(e_{i}) = e_{i},  Θ(f_{i}) = f_{i} for 1 ≤ i ≤ n
Θ(e_{n+1}) = −e_{n+1},  Θ(f_{n+1}) = −f_{n+1}.

We furthermore use the bases (e′_{i})_{1≤i≤n} of X^{∗}(T_{H}) and (f′_{i})_{1≤i≤n} of X_{∗}(T_{H}) from example 1.13. It is clear that we have an isomorphism ι : X^{∗}(T_{H}) ≃ X^{∗}(T)^{Θ} given by e′_{i} ↦ e_{i} and a dual isomorphism P_{Θ} : (X_{∗}(T)_{Θ})_{free} ≃ X_{∗}(T_{H}) induced by the dual map P_{Θ} : f_{n+1} ↦ 0 and P_{Θ}(f_{i}) ↦ f′_{i} for 1 ≤ i ≤ n. It is clear that this P_{Θ} induces the map (xii). Recall the (co)root systems of SO_{2n+2}

Φ(SO_{2n+2}, T) = {±e_{i} ± e_{j} | 1 ≤ i < j ≤ n + 1}
Φ^{∨}(SO_{2n+2}, T) = {±f_{i} ± f_{j} | 1 ≤ i < j ≤ n + 1}.

We have P_{Θ}(f_{i} ± f_{j}) = f′_{i} ± f′_{j}, S′_{Θ}(e_{i} ± e_{j}) = ι(e′_{i} ± e′_{j}) for 1 ≤ i < j ≤ n and P_{Θ}(f_{i} ±f_{n+1}) = f′_{i}, S′_{Θ}(e_{i} ±e_{n+1}) = e_{i} ±e_{n+1} +Θ(e_{i} ±e_{n+1}) = ι(2·e′_{i}) for 1 ≤ i ≤ n. The relations (v) and (vi) now follow from the knowledge of the (co)root system of Sp_{2n}. It is clear that S′_{Θ} maps the simple roots e_{i} − e_{i+1}, e_{n} ± e_{n+1} of SO_{2n+2} to the simple roots e′_{i}−e′_{i+1}, 2·e′_{n} of Sp_{2n} and P_{Θ} maps the simple coroots f_{i}−f_{i+1}, f_{n}±f_{n+1} of SO_{2n+2} to the simple coroots f′_{i} − f′_{i+1}, f′_{n} of Sp_{2n}.

2 Centralizers

Theorem 2.1 (Steinberg). Let T be a Θ–stable, maximal subtorus of G and t ∈ T (F̄). The group G^{tΘ} of fixed point in G under int(t)∘Θ is reductive. The root system of the connected component (G^{tΘ})^{∘} of 1, viewed as a subsystem of Φ_{Θ} = P_{Θ}(Φ(T, G)), which might be identified with Φ_{res} = {α|_{T^{Θ∘}} | α ∈ Φ}, is given by

Φ(G^{tΘ∘}, T^{Θ∘}) ≃ \{P_{Θ}(α) ∈ Φ_{Θ} | S_{Θ}(α)(t) = \{ \begin{matrix}1 & if \frac{1}{2}P_{Θ}(α) ∉ Φ_{Θ} \\ −1 & if \frac{1}{2}P_{Θ}(α) ∈ Φ_{Θ} \end{matrix} \} .

Proof: [St, 8.1]

(2.2) Define N_{Θ}(t) = ∏_{i=0}^{ord(Θ)−1} Θ^{i}(t). In case Φ(G) is of type A_{2n} or one Θ–Orbit of components of type A_{2n} the root system Φ(G^{tΘ∘}, T^{Θ∘}) in 2.1 is a maximal reduced subsystem of

{P_{Θ}(α) | α(N_{Θ}(t)) = 1}.

(2.3) For an irreducible root system Φ with basis Δ and θ ∈ Aut(Φ, Δ) we denote by α̃ ∈ Φ^{−} the negative root such that −c(α̃)S_{Θ}(α̃) = −S′_{Θ}(α̃) is the highest root in S′_{Θ}(Φ^{+}) with respect to the basis S′_{Θ}(Δ). To (Φ, Δ, Θ) we associate the extended Dynkin diagram

Δ_{ext}(Φ, Θ) := P_{Θ}(Δ) ∪ {c(α̃)P_{Θ}(α̃)}

Proposition 2.4 (Dynkin). Let G and Θ be as in 1.2.

(a) For every t ∈ T there exists w ∈ W^{Θ}, such that Φ((G^{w(t)Θ})^{∘}) has a basis in Δ_{ext}(Φ, Θ).

(b) Every proper subsystem of Δ_{ext}(Φ, Θ) occurs as Φ ((G^{tΘ})^{∘}) for some t ∈ T.

Proof: For a detailed proof we refer to [Bal][2.42]. We remark that in case Φ = A_{2n} the extended Dynkin diagram (P_{Θ}(Φ), Δ_{ext}(Φ, Θ)) looks like

• > ∘ ∘ ∘ ∘ ∘ ∘ > ∘

Therefore (a) will follow in the case G = SL_{2n+1} from the fact, proven in lemma 4.9 that the groups G^{tΘ} are isomorphic to groups of the form

G^{tΘ} = ( ⋆ ; × ; ... ; ⋆ \begin{matrix}∗ & ∗ \\ ∗ & ∗ \end{matrix} ⋆ ; ... ; × ; ⋆ ) ≃ Sp(2m_{k}) × Gl(m_{k−1}) × · · · · · · × Gl(m_{1}) × SO(2m_{0} + 1)

where (m_{k}, m_{k−1}, …, m_{1}, m_{0}) runs through all partitions of n with m_{k}, m_{0} ≥ 0 and m_{i} ≥ 1 for 0 < i < k. This means that the G^{tΘ} are of type C_{m_{k}} × A_{m_{k−1}−1} × ⋯ × A_{m_{1}−1} × B_{m_{0}} and the Δ(G^{tΘ}) are subsets of P_{Θ}(Δ) ∪ {−P_{Θ}(α^{+})}, where α^{+} = ε_{1} − ε_{2n+1} is the highest root for G.

(2.5) In the following table we list all simple root systems Φ = Φ(G), such that the semisimple (simply connected split) group G has an outer automorphism Θ, together with the root systems Φ(H) of the stable endoscopic groups H of (G, θ). The ordinary simple roots are marked by a ∘, the additional root α̃ by a •. We get six blocks, separated by double lines, which contain the following information:

Φ(G) Δ_{ext}(Φ(G), id)  θ  Δ_{ext}(Φ(G), θ) ; Φ(H)  θ = id Δ_{ext}(Φ(H), id)

Here Δ_{ext}(Φ(G), id) is arranged such that the θ-orbits of roots are in vertical order. We may think of Δ(Φ(G), θ) as a quotient diagram of Δ(Φ(G), id), but their seems to be no rule which describes the additional root c(α̃)P_{Θ}(α̃). To obtain Δ_{ext}(Φ(H), id) one observes at first that Φ(Ĝ) ≃ Φ(G), since G is of type ADE, then remarks Δ(Φ(Ĥ), id) = Δ(Φ(Ĝ), θ), reversing the arrows in this diagram one gets the diagram of Δ(Φ(H), id), which finally has to be extended to Δ_{ext}(Φ(H), id).

10 2. Centralizers Table 2.6. Δ_{ext}(Φ(G), θ) versus Δ_{ext}(Φ(H), id)

Φ(G)  Δ_{ext}(Φ, id)  θ  Δ_{ext}(Φ, θ)
A_{2n−1}_{n≥3}  • ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ Ord(θ) = 2  < • ∘ ∘ ∘ ∘ ∘ ∘ ∘
B_{n}_{n≥3}  θ = id  > • ∘ ∘ ∘ ∘ ∘ ∘ ∘
A_{3}  • ∘ ∘ ∘ ∘  Ord(θ) = 2  ∘ < ∘ > •
B_{2}  θ = id  ∘ > ∘ < •
A_{2n}_{n≥1}  • ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ Ord(θ) = 2  >  > > resp. if n = 1 • ∘ ∘ ∘ ∘ ∘ • ∘ ∘
C_{n}_{n≥1}  θ = id  • > ∘ ∘ ∘ ∘ < ∘
D_{n+1}_{n≥3}  • ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ Ord(θ) = 2 • < ∘ ∘ ∘ ∘ > ∘
C_{n}_{n≥3}  θ = id  • > ∘ ∘ ∘ ∘ < ∘
E_{6}  • ∘ ∘ ∘ ∘ ∘ ∘ ∘  Ord(θ) = 2  ∘ ∘ > ∘ ∘ •
F_{4}  θ = id  ∘ ∘ < ∘ ∘ •
D_{4}  • ∘ ∘ ∘ ∘ ∘  Ord(θ) = 3  ∘ > ∘ •
G_{2}  θ = id  ∘ < ∘ •

(2.7) Comparison of diagrams: By construction the diagrams Δ_{ext}(Φ(G), θ) and Δ_{ext}(Φ(H), id) are arranged in vertical order on the right hand side of each of the six blocks, and the corresponding spherical diagrams are obtained from each other by reversing the arrows. By inspection we see that the same statement holds for the extended diagrams with the exception that in case A_{2n} ↔ C_{n} there is no reversal of the arrow joining the additional root with the standard diagram.

A similar phenomenon appears when we consider centralizers: If sθ ∈ G(F̄)θ and σ ∈ H(F̄) match they can be assumed to lie in the diagonal tori such that: T (F̄) ∋ s ↦ σ ∈ T_{H}(F̄) = T_{Θ}(F̄). If we compute Φ(G^{sθ}) and Φ(H^{σ}) using 2.1 and 2.4(b) we see by inspection that they can be arranged in vertical order as subdiagrams (in the sense of of 2.4(b)) of the diagrams Δ_{ext}(Φ(G), θ) and Δ_{ext}(Φ(H), id) , but the oriented arrows get reversed with the exception mentioned above. If we write (G^{sΘ})_{ad} resp. (H^{σ})_{ad} as product of simple groups, we therefore get common factors of type A and D but factors of type B in general correspond to factors of type C and vice versa and factors of type G_{2} and F_{4} come in with reversed arrows.

(2.8) Our strategy to prove the fundamental lemma in the case of classical split groups now goes as follows: By the Kazhdan lemma 5.5 the (twisted) stable orbital integral of gθ ∈ G(F)θ in the group G̃ can be replaced by the ordinary stable orbital integral in the group G^{sθ} of the topologically unipotent part u. Now G^{sθ} resp. H^{σ} is isogenous to a product G_{+}^{sθ} × G_{∗}^{sθ} resp. H_{+}^{σ} × H_{∗}^{σ} , such that G_{+}^{sθ} is of type B or C, H_{+}^{σ} is the other of these two types and G_{∗}^{sθ} is isogenous to H_{∗}^{σ}. Similarly the stable orbital integral of some γ ∈ H(F), which matches with g, can be computed as the stable orbital integral of the topologically unipotent part v in the group H^{σ}, where the residually semisimple part σ of γ matches with sθ. Decomposing u = (u_{+}, u_{∗}) and v = (v_{+}, v_{∗}) we get that u_{∗} and v_{∗} coincide up to stable conjugation und up to powering, so they have matching stable orbital integrals. The fundamental lemma for gθ and γ will now follow if we can assume that the stable orbital integrals of u_{+} and v_{+} match, which is essentially the BC conjecture 5.3, stated already in the introduction.

(2.9) In case Φ = A_{2n} consider the exact sequence:

1 ⟶ μ_{2n+1} ⟶ SL_{2n+1} ⟶ PGL_{2n+1} ⟶ 1.

Since the involution Θ acts as −1 on μ_{2n+1} ≃ ℤ/(2n + 1)ℤ one gets the long exact sequence

1 ⟼ \underbrace{κ(G)^{Θ}}_{=1} ⟶ G^{tΘ} ⟶ G_{ad}^{tΘ} ⟶ \underbrace{H^{1}(⟨Θ⟩, κ(G))}_{=1} ⟶ ...

for every t ∈ Inn(G).

Example 2.10. The case Φ = A_{4} Consider s ∈ T (F̄), where T is the diagonal torus in G = PGL_{5}. There are seven 12 2. Centralizers possible types of groups G^{sΘ} nonisomorphic over F̄. (ζ_{n} denotes a (fixed) primitive n–th root of unity in F̄.) > > ∘ ∘ ∘ ∘ ∘ ∘ • ∘ ∘ s = 1 1. Case: G^{sΘ} = SO(5) = {g ∈ Sl(5) | ^{t}g ·J ·g = J} is of type Φ(G^{sΘ}) = B_{2} and π_{1}(Φ(G^{sΘ})) ≃ ℤ/2.

> > ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ • s = ( \begin{matrix}ζ_{4} & · & · & · & · \\ · & ζ_{4} & · & · & · \\ · & · & 1 & · & · \\ · & · & · & ζ_{4}^{−1} & · \\ · & · & · & · & ζ_{4}^{−1} \end{matrix} ) 2. Case:

G^{sΘ} = Sp(2 · 2) = {g ∈ Sl(5) | ^{t}g · J̃ · g = J̃}, where J̃ = J · Diag(−1, −1, 1, 1, 1), is of type Φ(G^{sΘ}) = C_{2} and π_{1}(Φ(G^{sΘ})) = 1.

> > ∘ ∘ ∘ ∘ ∘ ∘ ∘ • ∘ s = ( \begin{matrix}ζ_{4} & · & · & · & · \\ · & 1 & · & · & · \\ · & · & 1 & · & · \\ · & · & · & 1 & · \\ · & · & · & · & ζ_{4}^{−1} \end{matrix} ) 3. Case:

G^{sΘ} ≃ SO(3) × Sp(2), is of type Φ(G^{sΘ}) = B_{1} × C_{1} = A_{1}^{2} and π_{1}(Φ(G^{sΘ})) ≃ ℤ/2.

> > ∘ ∘ ∘ ∘ ∘ ∘ • • ∘ s = ( \begin{matrix}ζ_{8} & · & · & · & · \\ · & 1 & · & · & · \\ · & · & 1 & · & · \\ · & · & · & 1 & · \\ · & · & · & · & ζ_{8}^{−1} \end{matrix} ) 4. Case:

G^{sΘ} ≃ SO(3) × Gl(1)

> > ∘ ∘ ∘ ∘ ∘ ∘ • ∘ • s = ( \begin{matrix}ζ_{8} & · & · & · & · \\ · & ζ_{8} & · & · & · \\ · & · & 1 & · & · \\ · & · & · & ζ_{8}^{−1} & · \\ · & · & · & · & ζ_{8}^{−1} \end{matrix} ) 5. Case:

G^{sΘ} ≃ Gl(2)

> > ∘ ∘ ∘ ∘ ∘ ∘ ∘ • • s = ( \begin{matrix}ζ_{4} & · & · & · & · \\ · & ζ_{8} & · & · & · \\ · & · & 1 & · & · \\ · & · & · & ζ_{8}^{−1} & · \\ · & · & · & · & ζ_{4}^{−1} \end{matrix} ) 6. Case:

G^{sΘ} ≃ Sp(2) × Gl(1)

> > ∘ ∘ ∘ ∘ ∘ ∘ • • • s = ( \begin{matrix}ζ_{6} & · & · & · & · \\ · & ζ_{12} & · & · & · \\ · & · & 1 & · & · \\ · & · & · & ζ_{12}^{−1} & · \\ · & · & · & · & ζ_{6}^{−1} \end{matrix} ) 7. Case:

G^{sΘ} ≃ Gl(1) × Gl(1)

3 Topological Jordan decomposition

Definition 3.1. Let tJZ_{p} be the category, whose objects are topological groups, such that the neighborhood filter of 1 has a basis consisting of pro–p–groups, and whose morphisms are the continuous homomorphisms.

Definition 3.2. An element g of G ∈ Ob(tJZ_{p}) is called

• strongly compact, if g lies in a compact subgroup of G.

• topologically unipotent, if lim_{n→∞} g^{p^{n}} = 1.

• residually semisimple, if g is of a finite order, which is prime to p.

(3.3) For an element g it is equivalent to be topologically unipotent and to lie in a pro–p–subgroup of G. In the definition above one can replace the sequence (p^{n}) by an arbitrary sequence (p^{n_{k}})_{k} satisfying lim_{k→∞} n_{k} = ∞.

Example 3.4. Let F be a p–adic field and G̃ an affine linear algebraic group. Then G̃(F) in an Object of tJZ_{p}.

An element g ∈ G̃(F) in the example 3.4 is strongly compact, iff the set g^{ℤ} is bounded inside G̃(F). A third equivalent formulation is, that the eigenvalues of ρ(g) are units in F̄^{×} for one/for all faithful representation(s) ρ : G → GL(V).

Each pro–p–group U has a unique structure as a topological ℤ_{p}–module, which extends the canonical structure as ℤ–module (comp. [Hasse, §15.2]).

Lemma 3.5 (topological Jordan decomposition). Let G be an object of tJZ_{p}. Every strongly compact g ∈ G has a unique decomposition

g = g_{u} · g_{s} = g_{s} · g_{u} ,

where g_{u} ∈ G is topologically unipotent and g_{s} ∈ G is residually semisimple.

Proof: We consider the (abelian!) closure \overline{⟨g^{ℤ}⟩} of the abelian group ⟨g^{ℤ}⟩, which is contained in a compact subgroup of G and is therefore itself compact. Since G ∈ Ob(tJZ_{p}) there exists an open pro–p–subgroup of \overline{⟨g^{ℤ}⟩}. The set U of all topologically unipotent elements in \overline{⟨g^{ℤ}⟩} contains this open subgroup, is a group since \overline{⟨g^{ℤ}⟩} is abelian, and is therefore an open pro–p–subgroup U. The compactness of \overline{⟨g^{ℤ}⟩} implies that U has a finite index N in it, which has to be prime to p. Since U is a ℤ_{p}–module and N ∈ ℤ_{p}^{×} there exists a (topologically unipotent!) element g_{u} ∈ U such that g^{N} = g_{u}^{N}. Since g_{u}g = gg_{u} the element g_{s} = g · g_{u}^{−1} satisfies g_{s}^{N} = 1, i.e. is residually semisimple, and we get g = g_{s}g_{u} and g = g · g_{u} · g_{u}^{−1} = g_{u} · g · g_{u}^{−1} = g_{u}g_{s}, i.e the existence of the decomposition is proved.

If g = g′_{u}g′_{s} = g′_{s}g′_{u} is a second topological Jordan decomposition with (g′_{s})^{N′} = 1 we choose a p-power Q = p^{m} such that Q ≡ 1 mod NN′ and get lim_{α→∞} g^{Q^{α}} = lim_{α→∞}(g′_{u})^{Q^{α}} · (g′_{s})^{Q^{α}} = lim_{α→∞}(g′_{u})^{Q^{α}} · g′_{s} = g′_{s}, and by the same argument: lim_{α→∞} g^{Q^{α}} = g_{s}, i.e. g′_{s} = g_{s} and therefore also g_{u} = g′_{u} i.e. the uniqueness assertion.

Corollary 3.6 (Properties of the topological Jordan decomposition).

(1) Let g ∈ G be strongly compact, N ∈ ℕ be prime to p, such that g^{N} lies in some pro–p–group, and let Q be a p–power with Q ≡ 1 (mod N). Then we have

lim_{m→∞} g^{Q^{m}} = g_{s}.

(2) We have g_{u} ∈ G^{g_{s}} and G^{g} = Cent(g_{u}, G^{g_{s}}).

(3) Residually semisimple elements are semisimple.

(4) Let m be prime to p and u be topologically unipotent. Then there exists a unique topologically unipotent u_{1} such that

u_{1}^{m} = u.

(5) The topological Jordan decomposition is functorial in the following sense: The strongly compact (resp. the topologically unipotent, resp. the residually semisimple) elements define functors from tJZ_{p} to Set. For each morphism φ in tJZ_{p} we have φ((·)_{s}) = (φ(·))_{s} und φ((·)_{u}) = (φ(·))_{u}. Especially:

(5a) If H is a closed subgroup of G and g ∈ H, then also g_{s} and g_{u} are in H.

Corollary 3.7. Let F and G̃ be as in example 3.4 and assume furthermore that |π_{0}(G̃)| is prime to p. Each strongly compact element g ∈ G̃(F) has a unique topological Jordan decomposition:

g = g_{u} · g_{s} = g_{s} · g_{u} ,

where g_{s} ∈ G̃(F) is residually semisimple and g_{u} ∈ (G̃)^{∘}(F) topologically unipotent.

The functoriality implies the following statements:

(1) Let ρ : G̃ → G̃′ be a morphism of (not necessarily connected) reductive groups, defined over a finite extension of F. Then we have ρ(g)_{s} = ρ(g_{s}) and ρ(g)_{u} = ρ(g_{u}).

(2) If g ∈ G̃(O_{F}), then the image of the topological Jordan decomposition under the reduction map is the Jordan decomposition in G̃(𝔽_{q}).

(Topologically unipotent elements must lie in (G̃)^{∘} since by assumption p does not divide |π_{0}(G̃)|.)

4 Classification of Θ-conjugacy classes

(4.1) If (G, Θ) is as in the examples 1.8 or 1.9, the problem of determining the Θ-conjugacy classes of elements sΘ ∈ G̃(F) is equivalent to determine the classes of h = sJ under the transformations h ↦ g · h ·^{t}g. Namely we have the following commutative diagram:

GL_{2n} ⟶^{h↦hJ^{−1}Θ} G̃L_{2n}
h↦g·h·^{t}g↓  ↓^{x↦g·x·g^{−1}}
GL_{2n} ⟶_{h↦hJ^{−1}Θ} G̃L_{2n}

If we decompose h = q + p in its symplectic part p and its symmetric part q we thus have to consider the problem of simultaneous normal forms for a symplectic and a symmetric bilinear form. To obtain results for orbital integrals we have to deal with this problem also over the ring of integers O_{F}. The problem can be attacked if we assume sΘ to be semisimple (resp. residually semisimple if we work over O_{F}).

(4.2) Notations: In the following R denotes either a field of characteristic 0 or the ring of integers O_{F} of a local p-adic field F, where p ≠ 2. We denote by 𝔪 the maximal ideal of R (i.e. 𝔪 = (0) if R is a field) and by κ = R/𝔪 the residue field in the case R = O_{F}.

Let M denote a free R-module of finite rank r with basis (b_{i})_{1≤i≤r}. A bilinear form q : M ×M → R is called unimodular if Δ(q) := det(q(b_{i}, b_{j})) ∈ R^{×}. This definition is obviously independent of the chosen basis (b_{i}) since Δ(q) is an invariant in R/(R^{×})^{2}. For h ∈ GL_{n}(R) we have the following bilinear forms b_{h} and b′_{h} on the module M = R^{n} of column vectors: b_{h}(m_{1}, m_{2}) = ^{t}m_{1} · h · m_{2} and b′_{h}(m_{1}, m_{2}) = ^{t}m_{1} ·^{t}h · m_{2}.

An element g ∈ GL(M) is called R-semisimple iff

• g is semisimple in the case R is a field,

• g is residually semisimple (i.e. has finite order prime to char(κ)) in the case R = O_{F}.

For h ∈ GL_{n}(R) we call N(h) = h ·^{t}h^{−1} the (right) norm of h and N_{l}(h) = ^{t}h^{−1} · h the left norm of h. N(h) and N_{l}(h) are conjugate by h in GL_{n}(R). Then h is called R-Θ-semisimple if N(h) (or equivalently N_{l}(h)) is R-semisimple.

We remark that h is R-Θ-semisimple if and only if h·J^{−1}·Θ is semisimple respectively residually semisimple as an element of GL_{n}(R) ⋊ ⟨Θ⟩.

Lemma 4.3. If p : M × M → R is a unimodular symplectic form, then there exists a basis (e_{1}, …, e_{g}, f_{g}, …, f_{1}) of M, such that p has standard form with respect to this basis, i.e. p(e_{i}, e_{j}) = p(f_{i}, f_{j}) = 0 and p(e_{i}, f_{j}) = δ_{ij}.

Proof: The standard procedure to get a symplectic basis of M applies for unimodular forms.

Lemma 4.4. If q : M × M → R is a unimodular symmetric bilinear form and R = O_{F}, then there exists a basis (e_{i})_{1≤i≤r} of M such that q(e_{i}, e_{j}) = δ_{ij} for (i, j) ≠ (r, r) and q(e_{r}, e_{r}) is some given element in the class of Δ(q) in R^{×}/(R^{×})^{2}.

Proof: Consider the reductions κ = R/𝔪, M̄ = M/𝔪M and q̄: M̄ ×M̄ → κ. Since quadratic forms over finite fields are classified by their discriminants, the analogous statement for q̄ holds. By lifting a basis from M̄ to M we can therefore assume that q(b_{i}, b_{j}) ≅ δ_{ij} mod 𝔪 for (i, j) ≠ (r, r). But now we can apply the Gram-Schmidt-Orthogonalization procedure (observe that elements congruent to 1 modulo 𝔪 are squares since p ≠ 2) to obtain the claim.

Lemma 4.5.

(a) If g ∈ GL(M) is R-semisimple then there exists a finite étale galois extension R′/R such that M′ = M ⊗_{R} R′ decomposes into the direct sum of eigenspaces: M′ = ⨁_{λ} M′_{λ}, where g acts on M′_{λ} as the scalar λ.

(b) If g = N_{l}(h) for an R-Θ-semisimple h ∈ GL_{n}(R) (see 4.2) then b_{h}(M′_{λ}, M′_{μ}) = 0 = b′_{h}(M′_{λ}, M′_{μ}) unless λμ = 1.

(c) The restrictions of the forms b_{h} and b′_{h} to M′_{1} and M′_{−1} are unimodular. For λ ≠ ±1 also the restrictions of b_{h}, b′_{h}, b_{h} + b′_{h} and b_{h} − b′_{h} to the modules N′_{λ} = M′_{λ} ⊕ M′_{λ^{−1}} are unimodular.

Proof: (a) The minimal polynomial χ(X) of g decomposes in pairwise different linear factors χ(X) = ∏_{i=1}^{r}(X−λ_{i}) over some extension ring of R. The ring R′ = R[λ_{i}]_{1≤i≤r} is finite étale and galois over R, since the λ_{i} are roots of unity of order prime to char(κ) in the case R = O_{F}. By the same reason we have

λ_{i} − λ_{j} ∈ (R′)^{×} for i ≠ j (i)

in both cases for R. We remark for later use that this statement remains correct if we add ±1 to the set of the λ_{i} (if they are not already among them). Therefore χ_{i}(X) = ∏_{j≠i}((X −λ_{j})·(λ_{i}−λ_{j})^{−1}) ∈ R′[X]. We have ∑_{i=1}^{r} χ_{i}(X) = 1 since the left hand side is a polynomial of degree r − 1 which has the value 1 at r different places. Therefore M′ is the sum of the subspaces M′_{λ_{i}} = χ_{i}(g)(M). Since (g − λ_{i}) · χ_{i}(g) equals χ(g) · ∏_{j≠i}(λ_{i} − λ_{j})^{−1} = 0, the spaces M′_{λ_{i}} are eigenspaces for g and the sum M′ = Σ_{i=1}^{r}M′_{λ_{i}} is direct.

(b) For m ∈ M′_{λ} and n ∈ M′_{μ} we have m = λ^{−1} · gm and n = μ · g^{−1}n. The claims follow immediately from the relations b_{h}(m, n) = λ^{−1} · b′_{h}(m, n) and

b_{h}(m, n) = μ · b′_{h}(m, n). (ii)

(c) In view of the orthogonality relations (b) and the unimodularity of h and ^{t}h the claims for the restrictions of b_{h} and b′_{h} follow immediately. By the formula (ii) above we have for m ∈ M_{λ}, n ∈ M_{λ^{−1}}:

(b_{h} ± b′_{h})(m, n) = (1 ± λ)b_{h}(m, n)
(b_{h} ± b′_{h})(n, m) = (1 ± λ^{−1})b_{h}(n, m).

Since 1 ± λ, 1 ± λ^{−1} ∈ (R′)^{×} by the remark following (i) above, the claim follows also for the restrictions of b_{h} ± b′_{h}.

Lemma 4.6. For an R-Θ-semisimple h ∈ GL_{n}(R) with decomposition h = p + q, where p is skew-symmetric and q symmetric, we have a direct sum decomposition for M = R^{n}

M = M_{+} ⊕ M_{−} ⊕ M_{0},

where M_{+} = ker p, M_{−} = ker q and M_{0} = (M_{+})^{⊥_{q}} ∩(M_{−})^{⊥_{p}} is the intersection of the orthogonal complement of M_{+} with the symplectic orthogonal complement of M_{−}. The restrictions

q_{+} = q | M_{+}×M_{+},  p_{−} = p | M_{−} × M_{−},
q_{0} = q | M_{0}×M_{0},  p_{0} = p | M_{0} × M_{0}

are unimodular.

Proof: We identify the matrices p, q ∈ GL_{n}(R) with the forms b_{p}, b_{q}. We take an extension R′/R as in Lemma 4.5(a) and compute

M′_{±1} = {m ∈ M′ | ^{t}h^{−1} · h · m = ±m} = {m ∈ M′ | hm = ±^{t}hm} = ker(h ∓^{t}h).

This means M′_{1} = ker(p | M′) and M′_{−1} = ker(q | M′) and implies M′_{1} = M_{+} ⊗_{R} R′, M′_{−1} = M_{−} ⊗_{R} R′. Since unimodularity can be checked after the extension R′/R and b_{h} restricts to q_{+} resp. p_{−} on M_{+} resp. M_{−}, we conclude from Lemma 4.5(c) that q_{+} and p_{−} are unimodular. Then it is clear that we have the claimed decomposition in (orthogonal and symplectic orthogonal) direct summands. By Lemma 4.5(b) we get M_{0}⊗_{R}R′ = ⨁_{λ≠±1} M′_{λ}. By Lemma 4.5(c) again we conclude that the restrictions of b_{h} + b′_{h} = 2q and b_{h} − b′_{h} = 2p to this module are unimodular. So p_{0} and q_{0} are unimodular.

Lemma 4.7 (Cayley transformation). Let p ∈ GL_{n}(R) be a skew-symmetric matrix. Let Sym_{n}(R)_{p−ess} denote the set of symmetric matrices q such that q ± p ∈ GL_{n}(R) and Sp(p, R)_{ess} the set of symplectic transformations b such that b − 1 ∈ GL_{n}(R). Then the following holds:

(a) We have a bijection

C : Sym_{n}(R)_{p−ess} → Sp(p, R)_{ess}, q ↦ (q − p)^{−1} · (q + p) = N_{l}(p + q).

The inverse map is C^{−1} : b ↦ p · (b + 1) · (b − 1)^{−1}.

(b) C induces a bijection between those elements q of Sym_{n}(R)_{p−ess}, for which p + q is R-Θ-semisimple, and the R-semisimple elements of Sp(p, R)_{ess}.

(c) The map C satisfies C(^{t}g · q · g) = g^{−1} · C(q) · g for g ∈ Sp(p, R).

Proof: (a) For q ∈ Sym_{n}(R)_{p−ess} we put h = p + q and b = ^{t}h^{−1} · h. We have

tb · h · b = ^{t}h · h^{−1} · h ·^{t}h^{−1} · h = ^{t}h ·^{t}h^{−1} · h i.e.
(iii)  ^{t}b · h · b = h and by transposing
(iv)  ^{t}b ·^{t}h · b = ^{t}h.

Subtracting the last two equations we get ^{t}b·p·b = p, i.e. b ∈ Sp(p, R). Furthermore b − 1 = (q − p)^{−1} · ((p + q) − (q − p)) = (q − p)^{−1} · 2p ∈ GL_{n}(R) by the assumptions. The map C is therefore defined. Conversely we get for b ∈ Sp(p, R)_{ess} and q = p · (b + 1) · (b − 1)^{−1} the equivalences:

q = ^{t}q ⇔ p · (b + 1) · (b − 1)^{−1} = ^{t}(b − 1)^{−1} ·^{t}(b + 1) · (−p)
⇔ (^{t}b − 1)p(b + 1) = (^{t}b + 1)p(1 − b)
⇔ ^{t}bpb +^{t}bp − pb − p = −^{t}bpb +^{t}bp − pb + p
⇔ ^{t}bpb = p ⇔ b ∈ Sp(p, R).

Furthermore q±p = p·((b + 1) ± (b − 1))·(b−1)^{−1} ∈ GL_{n}(R) since (b−1)^{−1}, 2b, 2, p ∈ GL_{n}(R). Therefore the map C^{−1} is also well defined. An easy calculation (as in the case of the usual Cayley transform) shows that the maps C and C^{−1} are inverse to another in their domain of definition.

(b) Since C(q) = N_{l}(p + q) = (p + q)^{−1} · N(p + q) · (p + q) this follows from the definition of R-Θ-semisimplicity.

(c) We have C(^{t}g · q · g) = (^{t}gqg − p)^{−1}(^{t}gqg + p) = g^{−1}(q − p)^{t}g^{−1} ·^{t}g(q + p)g = g^{−1} · (q − p)^{−1} · (q + p) · g = g^{−1} · C(q) · g for g ∈ Sp(p, R).

Lemma 4.8. If p is a unimodular symplectic form on a free R-module N and b ∈ Sp(p, R) is R-semisimple then there exists a b-invariant and with respect to p orthogonal direct sum decomposition N = N_{1} ⊕ N_{∗} such that b acts as identity on N_{1} and b|N_{∗} ∈ Sp(p_{∗}, R)_{ess}, where p_{∗} is the restriction of p to N_{∗}.

Proof: We argue as before: By lemma 4.5(a) we have for some finite étale ring extension R′/R a decomposition of N′ = N ⊗_{R} R′ into eigenspaces of b: N′ = ⨁ N′_{λ}, where b acts as the scalar λ on N′_{λ}. As in lemma 4.5(b) we can see, that p(N′_{λ}, N′_{μ}) = 0 unless λ · μ = 1. This implies that p is unimodular on N′_{1} and therefore on N_{1}, thus N is the direct sum of N_{1} and the p-orthogonal complement N_{∗} of N_{1}. Since b is a symplectic transformation, it leaves N_{∗} invariant. By the orthogonality relations for the N_{λ} we have N_{∗} ⊗_{R} R′ = ⨁_{λ≠1} N′_{λ}. Since λ − 1 ∈ (R′)^{×} for λ ≠ 1 the endomorphism b − 1 of N_{∗} induces an automorphism of N_{∗} ⊗_{R} R′ and is therefore itself an automorphism of N_{∗}, Lemma 4.9. Let h = p + q ∈ GL_{n}(R) be R-Θ-semisimple. Let G^{h,Θ}(R) = {g ∈ GL_{n}(R)|^{t}g · h · g = h}. Then the following holds:

(a) With the notations of lemma 4.6 and of lemma 4.7 we have

G^{h,Θ}(R) = O(q_{+}, R) × Sp(p_{−}, R) × (Sp(p_{0}, R) ∩ O(q_{0}, R))
≅ O(q_{+}, R) × (Sp(p_{−} ⊕ p_{0}, R) ∩ O(q_{−} ⊕ q_{0}, R))
≅ O(q_{+}, R) × Cent (C(q_{−} ⊕ q_{0}), Sp(p_{−} ⊕ p_{0}, R)) .

(b) In the situation and with the notations of lemma 4.5 we have moreover

(Sp(p_{0}, R′) ∩ O(q_{0}, R′)) = {(ϕ_{λ}) ∈ ∏_{λ≠±1} GL(M′_{λ})| ϕ_{λ^{−1}} = ^{t}ϕ_{λ} for all λ}

≅ ∏_{λ∈L} GL(M′_{λ})

where ϕ_{λ^{−1}} = ^{t}ϕ_{λ} means that b_{h}(ϕ_{λ^{−1}}m_{λ^{−1}}, ϕ_{λ}m_{λ}) = b_{h}(m_{λ^{−1}}, m_{λ}) for all m_{λ^{−1}} ∈ M′_{λ^{−1}}, m_{λ} ∈ M′_{λ} and where L denotes a subset of the set of all λ ≠ ±1, which takes from every pair {λ, λ^{−1}} exactly one member.

(c) (Sp(p_{−} ⊕ p_{0}) ∩ O(q_{−} ⊕ q_{0})) ≅ Cent (C(q_{−} ⊕ q_{0}), Sp(p_{−} ⊕ p_{0})) is a connected reductive smooth group scheme /R with connected special fiber, which becomes split over the finite étale extension R′/R.

(d) We have in the situation of 6.1

Cent(N (h), Sp_{2n}) ≅ Sp_{2(n−g)} × Cent (C(q_{−} ⊕ q_{0}), Sp(p_{−} ⊕ p_{0}))

where 2g is the rank of M_{−} ⊕ M_{0}.

(e) To obtain the intersections of G^{h,Θ}(R) with SL_{n}(R) one has only to replace O(q_{+}, R) by SO(q_{+}, R) on the right hand sides of (a).

Proof: (a) Since every g ∈ G^{h,Θ}(R) stabilizes the decomposition of lemma 4.6 one immediately gets the first two isomorphisms. The last one follows from lemma 4.7(c).

(b) Every g ∈ G^{h,Θ}(R) centralizes N_{l}(h) and therefore has to respect the decomposition of M_{0}⊗_{R}R′ in eigenspaces of N_{l}(h). The first description of Sp(p_{0}, R′)∩O(q_{0}, R′) follows now from 4.5(b). Since b_{h} is unimodular on M′_{λ^{−1}} ⊕M′_{λ} it induces an identification of M′_{λ^{−1}} with the dual space of M′_{λ}. This means that ϕ_{λ} can vary through the whole GL(M′_{λ}), while ϕ_{λ^{−1}} is then uniquely determined as the inverse of its adjoint. We remark that the condition ϕ_{λ} = ^{t}ϕ_{λ^{−1}} is equivalent to the condition ϕ_{λ^{−1}} = tϕ_{λ} and gives no extra restrictions. This is clear since we have b_{h}(m_{λ}, m_{λ^{−1}}) = b′_{h}(m_{λ^{−1}}, m_{λ}) = λ · b_{h}(m_{λ^{−1}}, m_{λ}) for m_{λ^{−1}} ∈ M′_{λ^{−1}}, m_{λ} ∈ M′_{λ} by (ii), so the two possible identifications of M′_{λ^{−1}} with the dual of M′_{λ} differ by a scalar and create the same adjoint. The last isomorphism follows.

(c) This follows from (a) and (b).

(d) This follows from the definition of N by the remark, that an element of Cent(b, Sp_{2n}(R)) has to respect the decomposition of lemma 4.8.

(e) is clear, since symplectic transformations have determinant 1.

Lemma 4.10. Let G/R = O_{F} be a connected reductive group with connected special fiber G ×_{O_{F}} κ and b ∈ G(R) be R-semisimple. If b′ = h_{F}^{−1} · b · h_{F} ∈ G(R) for some h_{F} ∈ G(F̄) then there exists h_{R} ∈ G(R) with b′ = h_{R}^{−1} · b · h_{R}.

Proof: This follows from [K3, Prop. 7.1.].

Lemma 4.11. Let R = O_{F} and h ∈ GL_{n}(R) be R-Θ-semisimple and h′ = ^{t}g_{F} · h · g_{F} ∈ GL_{n}(R) for some g_{F} ∈ GL_{n}(F̄). Then we have:

(a) If additionally det(g_{F}) ∈ F^{×} there exists g_{R} ∈ GL_{n}(R) with h′ = ^{t}g_{R} · h · g_{R}.

(b) If we only assume g_{F} ∈ GL_{n}(F̄) and if n is odd there exist g_{R} ∈ GL_{n}(R) and ϵ ∈ O_{F}^{×} such that h′ = ϵ ·^{t}g_{R} · h · g_{R}.

(c) We get the statement of (a) if we additionally assume that the discriminants of q_{+} and q′_{+} coincide in R^{×}/(R^{×})^{2}.

(d) Under the additional conditions h, h′ ∈ SL_{n}(O_{F}), g_{F} ∈ SL_{n}(F̄) and n odd we can find g_{R} ∈ SL_{n}(R) with h′ = ^{t}g_{R} · h · g_{R}.

Proof: We use the objects occurring in lemma 4.6 for h and denote the corresponding objects for h′ by a ′. We have rank(M_{+}) = dim(M_{+} ⊗_{R} F) = dim(M′_{+} ⊗_{R} F) = rank(M′_{+}). By transforming h and h′ with elements of GL_{n}(R) we can therefore assume (using lemma 4.3) that

M_{+} = M′_{+} = R^{m}, M_{0} ⊕ M_{−} = M′_{0} ⊕ M′_{−}, p_{∗} := p_{0} ⊕ p_{−} = p′_{0} ⊕ p′_{−}. (v)

The assumption and lemma 4.7(c) (applied in the case R = F̄) now imply that the elements C(0 ⊕ q_{0}) and C(0 ⊕ q′_{0}) of Sp(p_{∗}, R) are conjugate by an element of Sp(p_{∗}, F̄). By lemma 4.10 they are conjugate by an element g_{∗} ∈ Sp(p_{∗}, R), hence we get from lemma 4.7(c) the equality q′_{0} = ^{t}g_{∗} · q_{0} · g_{∗} and therefore p′_{0} + p′_{−} + q′_{0} = tg_{∗}(p_{0} + p_{−} + q_{0})g_{∗} in M_{0} ⊕ M_{−}. We have det(q′_{+}) = det(h′) · det(p′_{0} + p′_{−} + q′_{0})^{−1} = det(g_{F})^{2} · det(h) · det(p_{0} + p_{−} + q_{0})^{−1} = det(g_{F})^{2} · det(q_{+}) (observe det g_{∗} = 1).

If case (a) we conclude using R^{×} ∩ (F^{×})^{2} = (R^{×})^{2} and lemma 4.4, that q′_{+} and q_{+} are transformed via an element g_{+} ∈ GL_{n}(M_{+}), a statement which has been an additional assumption in (c) in view of lemma 4.4. We put g_{∗} and g_{+} together to g_{R} ∈ GL_{n}(R) which does the required job in cases (a) and (c).

We prove (b) for ϵ = det(q′_{+})/ det(q_{+}): We have h” := ϵ^{−1}h′ = ^{t}g′_{F} · h · g′_{F} for g′_{F} = \sqrt{ϵ^{−1}} · g_{F} ∈ GL_{n}(F̄). If 2r + 1 is the rank of M′_{+} we have det(q”_{+}) = det(q′_{+}) · ϵ^{2r+1} = det(q_{+}) · ϵ^{2r}. Thus the additional assumption of (c) is fulfilled and we get g_{R} ∈ GL_{n}(R) with h” = ^{t}g_{R} · h · g_{R}.

To prove (d) observe at first that we can assume the matrices transforming h and h′ into the standard form (v) being in SL_{n}(R) since one can modify them by elements of GL(M_{+}) and since rank(M_{+}) ≥ 1. From det(g_{F}) = 1 we furthermore get det(q′_{+}) = det(q_{+}) and therefore det(g_{+}) = ±1. Since we can replace g_{+} by −g_{+} if necessary and rank(M_{+}) is odd we can achieve det(g_{+}) = 1 and therefore det(g_{R}) = 1.

5 Orbital integrals

(5.1) For a (not necessarily connected) reductive group G̃/O_{F} with connected component G = G̃^{∘} and elements γ ∈ G̃(F), f ∈ C_{c}^{∞}(G̃(F)) we define the orbital integral by:

O_{γ}(f, G̃(F)) = ∫_{G(F)/G(F)^{γ}} f(xγx^{−1})dx/dx^{γ}

where G(F)^{γ} denotes the centralizer of γ in G(F) and where we have chosen Haar measures dx resp. dx^{γ} on G(F) resp. G(F)^{γ} such that

vol_{dx}(G(O_{F})) = 1 and vol_{dx^{γ}}((G^{γ})^{∘}(O_{F})) = 1.

If 1_{K} denotes the characteristic function of a compact open subset K ⊂ G(F), we will use the following abbreviation:

O_{γ}(1, G̃) = O_{γ}(1_{G̃(O_{F})}, G̃(F))

We further introduce stable orbital integrals

O_{γ}^{st}(f, G̃(F)) = ∑_{γ′∼γ} O_{γ′}(f, G̃(F))  respectively
O_{γ}^{st}(1, G̃) = ∑_{γ′∼γ} O_{γ′}(1_{G̃(O_{F})}, G̃(F))

where γ′ runs through a set of representatives for the conjugacy classes inside the stable conjugacy class of γ.

(5.2) Recall the construction of the quotient measure dg/dh on G/H for totally disconnected locally compact groups H ⊂ G, where G and H are unimodular (e.g. G and H are the sets of F -valued points of reductive groups). One defines

vol(KγH/H) = ∫_{G/H} 1_{KγH/H}(g)dg/dh := \frac{vol_{dg}(K)}{vol_{dh}(γ^{−1}Kγ ∩ H)},

where K ⊂ G is any open compact subgroup, and extends this by linearity to the space of all locally constant compactly supported functions on G/H.

Of course one has to prove a compatibility condition, if K′ ⊂ K is another open compact subgroup: For γ ∈ G let

KγH = ⋃_{j∈J}^{.} K′ · γ_{j} · γ · H

be a disjoint double coset decomposition with γ_{j} ∈ K. We have to prove:

vol(KγH/H) = ∑_{j∈J} vol(K′γ_{j}γH/H). (i)

Define C_{j} := {K′y ∈ K′\K | K′yγ ⊂ K′γ_{j}γH} ⊂ K′\K for j ∈ J. Then we have a disjoint decomposition

K′\K = ⋃_{j∈J}^{.} C_{j}  and isomorphisms
i_{j} : (γ^{−1}γ_{j}^{−1}K′γ_{j}γ ∩ H)\(γ^{−1}Kγ ∩ H) ⟶^{∼} C_{j}
h  ↦ K′ · γ_{j} · γ · h · γ^{−1}.

This implies

\frac{vol_{dg}(K)}{vol_{dg}(K′)} = [K : K′] = ∑_{j∈J}[(γ^{−1}Kγ ∩ H) : (γ^{−1}γ_{j}^{−1}K′γ_{j}γ ∩ H)]

= ∑_{j∈J} \frac{vol_{dh}(γ^{−1}Kγ ∩ H)}{vol_{dh}(γ^{−1}γ_{j}^{−1}K′γ_{j}γ ∩ H)}

or equivalently

\frac{vol_{dg}(K)}{vol_{dh}(γ^{−1}Kγ ∩ H)} = ∑_{j∈J} \frac{vol_{dg}(K′)}{vol_{dh}(γ^{−1}γ_{j}^{−1}K′γ_{j}γ ∩ H)} ,

which is the claim (i).

The crucial statement we need in the following is the following type of a fundamental lemma:

Conjecture 5.3. If the regular algebraically semisimple and topologically unipotent elements u ∈ SO_{2n+1}(F) and v ∈ Sp_{2n}(F) are BC-matching (see 1.12) then

O_{u}^{st}(1, SO_{2n+1}) = O_{v}^{st}(1, Sp_{2n}). (BC_{n})

The (easy) case (BC_{1}) is proved in [Fl1, Stable case I in Proof of Theorem]. The case (BC_{2}) is essentially proved in [Fl2, Part II], as will be explained in 7.9.

Warning: While (BC_{1}) is an immediate consequence of the exceptional isogeny i_{2} : Sp_{2} = SL_{2} ↠ PGL_{2} = SO_{3} and the fact, that γ^{2} and i_{2}(γ) are BC-matching for γ ∈ SL_{2}(F), the statement (BC_{2}) is much deeper, since the exceptional isogeny i_{4} : Sp_{4} ↠ SO_{5} does not satisfy the analogous matching property.

Remark 5.4. It follows immediately from the construction in 1.12 that we have a bijection between F -rational conjugacy classes in SO_{2n+1}(F̄) and in Sp_{2n}(F̄). By the theorem of Steinberg each F -rational conjugacy class in Sp_{2n}(F̄) contains a rational element, since Sp_{2n} is quasisplit and simply connected. But the same statement holds for F -rational topologically unipotent conjugacy classes in SO_{2n+1}(F̄) as well:

If u ∈ SO_{2n+1}(F̄) is topologically unipotent and represents an F -rational conjugacy class, consider its two preimages v_{1} and v_{2} = −v_{1} in Spin_{2n+1}. Since p ≠ 2 we have lim_{n→∞} v_{2}^{p^{n}} = − lim_{n→∞} v_{1}^{p^{n}}, so that exactly one of the elements v_{1}, v_{2} is topologically unipotent, say v_{1}. Since the Galois group respects the property to be topologically unipotent, the conjugacy class of v_{1} is F -rational and therefore contains an F - rational element v′ by the theorem of Steinberg. The image of v′ in SO_{2n+1}(F) is the desired F -rational representative of the conjugacy class of u. Thus to every topologically unipotent element in v ∈ Sp_{2n}(F) is associated at least one BC-matching u ∈ SO_{2n+1}(F) and vice versa.

Lemma 5.5 (Kazhdan-Lemma).

(a) For G̃ = G ⋊ ⟨Θ⟩ as in 1.2 let us assume that the following statement holds:

(∗) If s_{1}Θ and s_{2}Θ for s_{1}, s_{2} ∈ G(O_{F}) are residually semisimple and conjugate by an element of G(F) then they are also conjugate by an element of G(O_{F}).

If γΘ = u·sΘ = sΘ·u is a topological Jordan decomposition, where γ ∈ G(O_{F}), u is topologically unipotent and sΘ residually semisimple, we have

O_{γΘ}(1, G̃) = \frac{1}{[G^{sΘ}(O_{F}) : (G^{sΘ})^{∘}(O_{F})]} · O_{u}(1, G^{sΘ})

(b) Let H/O_{F} be connected reductive with connected special fiber. For h ∈ H(O_{F}) with topological Jordan decomposition h = v · b = b · v, where v is topologically unipotent and b residually semisimple, we have

O_{h}(1, H) = \frac{1}{[H^{b}(O_{F}) : (H^{b})^{∘}(O_{F})]} · O_{v}(1, H^{b})

Proof: (a) We first prove:

(**) We have gγΘg^{−1} ∈ G(O_{F})Θ if and only if g is of the form g = k · x where k ∈ G(O_{F}) and x ∈ G^{sΘ}(F) satisfies xux^{−1} ∈ G^{sΘ}(O_{F}).

The direction ”⇐” is easy: Under the hypothesis we have gγΘg^{−1} = kxusΘx^{−1}k^{−1} = k(xux^{−1})(sΘ)k^{−1} ∈ G(O_{F}). For the converse direction ”⇒” let us assume that gγΘg^{−1} ∈ G(O_{F})Θ. The topological Jordan decomposition is gγΘg^{−1} = (gug^{−1}) · (gsΘg^{−1}). Since ⟨G(O_{F}), Θ⟩ is a closed subgroup of G̃(F) we conclude from 3.6(5a) that gsΘg^{−1} ∈ G(O_{F})Θ and gug^{−1} ∈ G(O_{F}). By the first inclusion and assumption (∗) we get an element k ∈ G(O_{F}) such that g(sΘ)g^{−1} = k(sΘ)k^{−1}, which implies x = k^{−1} · g ∈ G^{sΘ}(F), where G^{sΘ} is the centralizer of sΘ in G. Using g = kx the inclusion gug^{−1} ∈ G(O_{F}) is now equivalent to xux^{−1} ∈ G(O_{F}), which proves (∗∗).

To finish the proof we introduce the double coset decomposition

{g ∈ G(F) | gγΘg^{−1} ∈ G(O_{F})Θ} = ⋃_{i∈I}^{.} G(O_{F}) · g_{i} · G^{γΘ},

where we can assume g_{i} ∈ G^{sΘ}(F) in view of (∗∗). Again from (∗∗) we get the double coset decomposition

{x ∈ G^{sΘ}(F) | xux^{−1} ∈ G^{sΘ}(O_{F})} = ⋃_{i∈I}^{.} G^{sΘ}(O_{F}) · g_{i} · G^{γΘ},

so that it remains to prove

(ii) ∑_{i∈I} \frac{vol_{dg}(G(O_{F}))}{vol_{dh}(g_{i}^{−1} · G(O_{F}) · g_{i} ∩ G^{γΘ}(F))}

= \frac{1}{[G^{sΘ}(O_{F}) : (G^{sΘ})^{∘}(O_{F})]} · ∑_{i∈I} \frac{vol_{dη}(G^{sΘ}(O_{F}))}{vol_{dh}(g_{i}^{−1} · G^{sΘ}(O_{F}) · g_{i} ∩ G^{γΘ}(F))} ,

where dη is a Haar measure on G^{sΘ}(F) satisfying vol_{dη}((G^{sΘ})^{∘}(O_{F})) = 1. This implies vol_{dη}(G^{sΘ}(O_{F})) = [G^{sΘ}(O_{F}) : (G^{sΘ})^{∘}(O_{F})]. On the other hand we claim

g_{i}^{−1} · G(O_{F}) · g_{i} ∩ G^{γΘ}(F) = g_{i}^{−1} · G^{sΘ}(O_{F}) · g_{i} ∩ G^{γΘ}(F). (iii)

The inclusion ”⊃” being trivial let us assume that g = g_{i}^{−1} · σ · g_{i} is an element of the left hand side. But g_{i} ∈ G^{sΘ}(F) and g ∈ G^{γΘ}(F) ⊂ G^{sΘ}(F) imply σ ∈ G^{sΘ}(F). Since G^{sΘ}(F) ∩ G(O_{F}) = G^{sΘ}(O_{F}) we get that g lies in the right hand side, i.e. (iii) is proved. (ii) now follows immediately.

(b) is now clear: We have G̃ = G i.e. Θ = 1 and the assumption (∗) is satisfied by lemma 4.10.

The following lemmas will be useful in later chapters.

Lemma 5.6. If N ∈ ℕ is prime to p then we have for a reductive group G/O_{F} and γ ∈ G(F)

O_{γ^{N}} (1, G) = O_{γ}(1, G).

Proof: Notice that g · γ · g^{−1} lies in the closure of (g · γ^{N} · g^{−1})^{ℤ} if N ∈ ℤ_{p}^{×}. This gives the equivalence g · γ^{N} · g^{−1} ∈ G(O_{F}) ⟺ g · γ · g^{−1} ∈ G(O_{F}), which implies the identity of orbital integrals.

Lemma 5.7. If G/O_{F} is of the form G = G_{1} × Z with a reductive group G_{1} and a finite group Z ≃ Z(O_{F}) then we have for γ ∈ G_{1}(F) ⊂ G(F) the following identity of orbital integrals:

O_{γ}(1, G) = O_{γ}(1, G_{1}).

Proof: We have G(F)/G(F)^{γ} ≃ G_{1}(F)/G_{1}(F)^{γ} since Z ⊂ G^{γ}, and the normalized Haar-measures on G_{1}(F) and G_{1}^{γ}(F) are the restrictions of the normalized Haarmeasures on G(F) and G^{γ}(F), since G^{∘} = G_{1}^{∘} and (G^{γ})^{∘} = (G_{1}^{γ})^{∘}. The claim follows.

Lemma 5.8. Let 1 → T → G → H → 1 be an exact sequence of algebraic groups over O_{F} where T is a split torus. Then we have for γ ∈ G(F) with image η ∈ H(F):

O_{γ}^{st}(1, G) = O_{η}^{st}(1, H).

Proof: We use the fact that the image of (G^{γ})^{∘} in H is (H^{η})^{∘}. By Hilbert 90 we get exact sequences 1 → T (F) → G(F) → H(F) → 1 and 1 → T (F) → (G^{γ})^{∘}(F) → (H^{η})^{∘}(F) → 1, so that we have an isomorphism G(F)/(G^{γ})^{∘}(F) ≃ H(F)/(H^{η})^{∘}(F). Since (H^{η})^{∘} has finite index in H^{η} we can compute O_{η}(1, H) as ∫_{H(F)/(H^{η})^{∘}(F)} 1_{H(O_{F})}(hηh^{−1})dh/dh^{η}. Similarly

O_{γ}(1, G) = ∫_{G(F)/(G^{γ})^{∘}(F)} 1_{G(O_{F})}(gγg^{−1})dg/dg^{γ}.

Now the quotient measures on G(F)/(G^{γ})^{∘}(F) and H(F)/(H^{η})^{∘}(F) coincide since G(O_{F}) ↠ H(O_{F}) and (G^{γ})^{∘}(O_{F}) ↠ (H^{η})^{∘}(O_{F}), and we conclude O_{γ}(1, G) = O_{η}(1, H).

It remains to check that the set St_{γ} of conjugacy classes inside the stable conjugacy class of γ maps bijectively to the corresponding set St_{η} associated to η. But in the following commutative diagram of abelian groups with exact rows and columns the map ι must be an isomorphism:

H^{1}(F, T)  H^{1}(F, T)
↓  ↓
1 ⟶ St_{γ} ⟶ H_{ab}^{1}(F, G^{γ}) ⟶ H_{ab}^{1}(F, G)
ι↓  ↓  ↓
1 ⟶ St_{η} ⟶ H_{ab}^{1}(F, H^{γ}) ⟶ H_{ab}^{1}(F, H)
↓  ↓
H^{2}(F, T)  H^{2}(F, T).

Here H_{ab}^{1}(F, .) denotes the abelianized cohomology of [Boro] which coincides for nonarchimedean F as a pointed set with the usual cohomology.

28 6. Comparison between PGL_{2n+1} and Sp_{2n}

6 Comparison between PGL_{2n+1} and Sp_{2n}

Recall (see 4.2) that R is either a field of characteristic 0 or the ring of integers O_{F} of a local non archimedean field F with residue characteristic ≠ 2.

(6.1) The explicit norm map N. Our final goal being the comparison of Θtwisted stable orbital integrals on PGL_{2n+1} with stable orbital integrals on Sp_{2n}, we will represent elements of PGL_{2n+1} by elements of the groups GL_{2n+1} resp. SL_{2n+1}. Let GL_{n}(R)_{RΘss}/traf resp. SL_{n}(R)_{RΘss}/traf be the set of transformation classes of R-Θ-semisimple (see 4.2) elements of h ∈ GL_{n}(R) resp. h ∈ SL_{n}(R) under the transformations h ↦ ^{t}ghg for g ∈ GL_{n}(R) resp g ∈ SL_{n}(R). Similarly let Sp_{2g}(R)_{Rss}/conj be the set of conjugacy classes of R-semisimple elements in Sp_{2g}(R). We define a norm map

N : GL_{2n+1}(R)_{RΘss}/traf ⟶ Sp_{2n}(R)_{Rss}/conj

as follows: If h = p + q ∈ GL_{2n+1}(R) represents a class of the left hand side, we decompose M = R^{2n+1} = M_{+} ⊕ M_{−} ⊕ M_{0} as in lemma 4.6. We consider M_{+} as the degenerate part of M with respect to p and denote the non degenerate part by M_{∗} := M_{−} ⊕ M_{0}. Since p_{∗} = p_{−} ⊕ p_{0} is a unimodular form on M_{∗} we can find a basis (e_{1}, …, e_{g}, f_{g}, …, f_{1}) of M_{∗} such that p_{∗} has standard form with respect to this basis by lemma 4.3. Let P_{∗} resp. Q_{∗} be the matrix describing the (skew-) symmetric bilinear form p_{∗} resp. q_{−} ⊕ q_{0} with respect to this basis (q_{−} is the zero form). Thus P_{∗} = J_{2g} and Sp(P_{∗}) = Sp_{2g}. Now N (h) or more precisely the image of the class of h under the norm map N is defined to be the Sp_{2n}(R)-conjugacy class of 1_{2(n−g)} × C(Q_{∗}) ∈ Sp_{2(n−g)}(R) × Sp_{2g}(R) ⊂ Sp_{2n}(R), where we use the Cayley-transform-map C from lemma 4.7.

Remark 6.2. In the situation where the decomposition M = R^{2n+1} = M_{+}⊕M_{∗} is of the form M = R^{2(n−g)+1}⊕R^{2g} the matrix h splits into the blocks h_{+} ∈ GL_{2(n−g)+1}(R) and h_{∗} ∈ GL_{2g}(R) so that N_{l}(h_{∗}) = ^{t}h_{∗}^{−1} · h_{∗} is a symplectic transformation with respect to the alternating part p_{∗} of h_{∗}. Then C(Q_{∗}) ∈ Sp_{2g}(R) is the conjugate of N_{l}(h_{∗}) by a matrix, which transforms p_{∗} into the standard form J_{2g}.

Proposition 6.3. Let R be as above. Then the following statements hold:

(a) The map N : GL_{2n+1}(R)_{RΘss}/traf ⟶ Sp_{2n}(R)_{Rss}/conj is well defined and surjective. In the case R = O_{F} its fibers are of order 2 = #(R^{×}/(R^{×})^{2}) and describe the two different classes of unimodular quadratic forms on M_{+}.

(b) The restriction N_{SL} of N to SL_{2n+1}(R)_{RΘss}/traf is surjective as well. It is bijective if R is an algebraically closed field or if R = O_{F}.

(c) If h represents a class in GL_{2n+1}(R)_{RΘss}/traf then the image of h · J^{−1}Θ in PGL_{2n+1}(R) ⋊ ⟨Θ⟩ matches with N (h) in the sense of Θ-endoscopy.

Proof: (a) and (b) The choices made in constructing N (h) only allow Q_{∗} to be replaced by some ^{t}g · Q_{∗} · g for g ∈ Sp(P_{∗}, R). By lemma 4.7(c) this does not change the conjugacy class of N (h). Therefore the map N is well defined. To prove surjectivity first observe that each class in Sp_{2n}(R)_{Rss}/conj has a representative of the form (1_{2(n−g)}, b) with b ∈ Sp_{2g}(R)_{ess} by lemma 4.8 with a unique g ≤ n. The Sp_{2g}(R)-conjugacy-class of b is unique. The bijectivity of the Cayley-transform map and property 4.7(c) then imply that there is a Q_{∗} ∈ Sym_{2g}(R), which is unique up to transformations with elements of Sp_{2g}(R) = Sp(P_{∗}, R), such that b = C(Q_{∗}). Now we consider the unimodular bilinear form h_{∗} = P_{∗} + Q_{∗} on R^{2g} and some unimodular symmetric bilinear form q_{+} on R^{2(n−g)+1}. The form q_{+} ⊕ h_{∗} on R^{2n+1} is then unimodular and R-Θ-semisimple. Since we can choose q_{+} in such a way that det(q_{+} ⊕ h′_{∗}) = 1 we get the surjectivity statements of (a) and (b). Since the transformation class of h′_{∗} is unique by the considerations above and since h = q_{+} ⊕ h_{∗} we conclude that the fibers of N correspond to the transformation classes of unimodular quadratic forms on M_{+}. The remaining statements of (a) and (b) now follow from lemma 4.4.

(c) By the definition of matching (1.11) we can work over R = F̄ and therefore may assume that γ = h · J_{2n+1}^{−1} has diagonal form γ = diag(t_{1}, …, t_{2n+1}). After applying a permutation in W_{SO_{2n+1}} we may assume

(i) t_{i} ≠ t_{2n+2−i} for i ≤ g and t_{i} = t_{2n+2−i} for g + 1 ≤ i ≤ 2n + 1 − g.

We have:

h = antidiag(t_{1}, −t_{2}, t_{3}, …, t_{2n+1})
h ±^{t}h = antidiag(t_{1} ± t_{2n+1}, −t_{2} ∓ t_{2n}, t_{3} ± t_{2n−1}, …, t_{2n+1} ± t_{1})
th^{−1} · h = diag(t_{2n+1}/t_{1}, t_{2n}/t_{2}, …, t_{n+2}/t_{n}, 1, t_{n}/t_{n+2}, …, t_{1}/t_{2n+1})

This means that M_{+} ≃ R^{2(n−g)+1} is spanned by the standard basis elements e_{g+1}, …, e_{2n+1−g} of R^{2n+1}, and M_{∗} = M_{−} ⊕ M_{0} by e_{1}, …, e_{g}, e_{2n+2−g}, …, e_{2n+1}. Since h −^{t}h is an antidiagonal matrix, its non degenerate part can be transformed by a diagonal matrix d into the standard form J_{2g}. Now we use remark 6.2 to get the following representative for N (h), observing that conjugation by d does not change a diagonal matrix:

diag(t_{2n+1}/t_{1}, t_{2n}/t_{2}, …, t_{n+2}/t_{n}, t_{n}/t_{n+2}, …, t_{1}/t_{2n+1}),

which may be conjugated by an element of the Weylgroup into the form

diag(t_{1}/t_{2n+1}, t_{2}/t_{2n}, …, t_{n}/t_{n+2}, t_{n+2}/t_{n}, …, t_{2n+1}/t_{1}).

The claim now follows from example 1.13.

Corollary 6.4. For every semisimple γ̄Θ ∈ { \atop PGL}_{2n+1}(F) there exists a semisimple η ∈ Sp_{2n}(F) matching with η in the sense of 1.11 and vice versa.

Proof: If γ ∈ GL_{2n+1}(F) represents a given γ̄ one applies part (c) of the proposition to h = γ · J_{2n+1}. If η is given one applies (b) and (c).

Proposition 6.5. Let Z = Cent(GL_{2n+1}) ≃ 𝔾_{m} denote the center of GL_{2n+1}. Let γ̄ ∈ PGL_{2n+1}(F) be represented by γ ∈ GL_{2n+1}(F). Since 2n + 1 is odd we can achieve that det(γ) has even valuation. Then

(ii) O_{γ̄Θ}(1, { \atop PGL}_{2n+1}) = 2 · O_{γΘ}(1, G̃L_{2n+1}).

If moreover γΘ is strongly compact with topological Jordan decomposition γΘ = u · (sΘ) = (sΘ) · u we have u ∈ SL_{2n+1}(F) and get

(iii) O_{γ̄Θ}(1, { \atop PGL}_{2n+1}) = O_{u}(1, SL_{2n+1}^{sΘ})

Proof: The relation ḡ · γ̄Θ · ḡ^{−1} ∈ \begin{matrix} \\ PGL & 2n+1 \end{matrix}(O_{F}) means g · γ · Θ(g)^{−1} = ζ · k with ζ ∈ Z(F) ≃ F^{∗} and k ∈ GL_{2n+1}(O_{F}). Since det(Θ(g)) = det(g)^{−1} the relation implies taking determinants

(iv) det(g)^{2} · det(γ) · ζ^{−2n−1} ∈ O_{F}^{∗} .

This implies that ζ has even valuation 2m for m ∈ ℤ, since the valuation of det(γ) was assumed to be even. If we replace g by g′ = ζ_{O} · ϖ^{−m} · g for ζ_{O} ∈ O_{F}^{∗} we get g′ · γ · Θ(g′)^{−1} ∈ GL_{2n+1}(O_{F}). Conversely the equation (iv) implies that every g′ ∈ g · Z(F) with this property must be of the stated form. Next observe that the condition ḡ ∈ PGL_{2n+1}^{γΘ}(F) means that we have for some representative g ∈ GL_{2n+1}(F) of ḡ and some ζ ∈ Z(F) ≃ F^{∗} the relation gγΘ(g)^{−1} = ζγ. This implies the determinant equation: det(g)^{2} = ζ^{2n+1}. Putting ρ = det(g)/ζ^{n} ∈ Z(F) this implies ζ = ρ^{2} and det(g) = ρ^{2n+1}. If we replace g by ρ^{−1} · g we get gγΘ(g)^{−1} = γ and det(g) = 1 . The only other element in g · Z(F) having the first property is −g, but det(−g) = −1. This means that we have isomorphisms

GL_{2n+1}(F)^{γΘ} ⟶^{∼} SL_{2n+1}(F)^{γΘ} × {±1}  and
SL_{2n+1}(F)^{γΘ} ⟶^{∼} PGL_{2n+1}^{γ̄Θ}(F).

Since the normalized Haar measure on PGL_{2n+1}(F) is the quotient of the normalized Haar measure on GL_{2n+1}(F) by the normalized Haar measure on Z(F) (i.e. vol(Z(O_{F})) = 1) and since the normalized measure on GL_{2n+1}(F)^{γΘ} restricts to the normalized Haar measure on PGL_{2n+1}^{γ̄Θ}(F) ≃ SL_{2n+1}^{γΘ}(F), the above considerations imply the relation (ii). If γΘ is strongly compact we can assume that γ ∈ GL_{2n+1}(O_{F}) and apply lemma 5.5 to get

O_{γ̄Θ}(1, { \atop PGL}_{2n+1}) = O_{u}(1, GL_{2n+1}^{sΘ})

observing that [GL_{2n+1}^{sΘ}(O_{F}) : (GL_{2n+1}^{sΘ})^{∘}(O_{F})] = 2. But since GL_{2n+1}(F)^{sΘ} ≃ SL_{2n+1}(F)^{sΘ} × {±1} we can apply lemma 5.7 to conclude (iii).

Lemma 6.6. Let h = sJ ∈ GL_{2n+1}(O_{F}) be R-Θ-semisimple and b = (1_{2(n−g)}, b_{∗}) ∈ Sp_{2n}(O_{F}) a representing element of N (h) with b_{∗} ∈ Sp_{2g}(O_{F})_{ess}. Since M_{+} is of odd rank 2(n − g) + 1 we can identify (M_{+}, q_{+}) with (O_{F}^{2(n−g)+1}, ϵq_{sp}) for some ϵ ∈ O_{F}^{×} and the standard splitform q_{sp}. Assume that we have BC-matching algebraically semisimple and topologically unipotent elements

u_{+} ∈ SO_{2(n−g)+1}(F) ≃ SO(q_{+}) and v_{+} ∈ Sp_{2(n−g)}(F) ≃ ker(b − 1)(F) ∩ Sp_{2n}(F)

and an additional algebraically semisimple and topologically unipotent element

u_{∗} ∈ SO(q_{∗})(F) ∩ Sp(p_{∗})(F) ≃ Cent(b_{∗}, Sp_{2g}(F)).

Then the elements γΘ = sΘ · (u_{+}, u_{∗}) = (u_{+}, u_{∗}) · sΘ ∈ PGL_{2n+1}(F)Θ and η := (v_{+}^{2}, u_{∗}^{2}) · b = b · (v_{+}^{2}, u_{∗}^{2}) ∈ Sp_{2n}(F) match.

Proof: As in the proof of lemma 6.1(c) we work in the case F = F̄ and assume that γ resp. η lie in the diagonal tori. The same holds for the residually semisimple parts s resp. b and the topologically unipotent parts u = (u_{+}, u_{∗}) and v = (v_{+}^{2}, u_{∗}^{2}). As the matching of sΘ and b is already proved in 6.1(c) we only have to examine the topologically unipotent elements. We can make the assumption (i) and write

u_{+} = diag(w_{g+1}, …, w_{n}, 1, w_{n}^{−1}, …, w_{g+1}^{−1}) ∈ SO_{2(n−g)+1}(F̄)
u_{∗} = diag(w_{1}, …, w_{g}, w_{g}^{−1}, …, w_{1}^{−1}) ∈ Cent(b_{∗}, Sp_{2g}(F̄))

By the definition of BC-matching we can assume

v_{+} = diag(w_{g+1}, …, w_{n}, w_{n}^{−1}, …, w_{g+1}^{−1}) ∈ Sp_{2(n−g)}(F̄)

Taking everything together we get from the description of M_{+} and M_{∗} in the proof of lemma 6.1(c):

u = (w_{1}, …, w_{n}, 1, w_{n}^{−1}, …, w_{1}^{−1})
v = (w_{1}^{2}, …, w_{n}^{2}, w_{n}^{−2}, …, w_{1}^{−2})

and the claim follows again from example 1.13.

The statement of the following theorem is the fundamental lemma for semisimple elements in the stable endoscopic situation (Sp_{2n}, { \atop PGL}_{2n+1}). Recall that the fundamental lemma also predicts the vanishing of orbital integrals for those rational elements, which match with no rational elements on the other side. But in view of corollary 6.4 this case does not occur.

Theorem 6.7. If the semisimple elements γ̄Θ ∈ { \atop PGL}_{2n+1}(F) and η ∈ Sp_{2n}(F) match in the sense of 1.11 and if conjecture (BC_{m}) is true for all m ≤ n then we have

O_{γ̄Θ}^{st}(1, { \atop PGL}_{2n+1}) = O_{η}^{st}(1, Sp_{2n}). (v)

Proof: \underline{Step 1} (Reductions): In the first step we will prove that the nonvanishing of one side of (v) implies the nonvanishing of the other side and that we can reduce to the following situation:

• γ ∈ GL_{2n+1}(O_{F})

• η ∈ Sp_{2n}(O_{F})

• the topological Jordan decompositions are of the form

γΘ = (u_{+}, u_{∗}) · sΘ and η = (v_{+}^{2}, v_{∗}^{2}) · b such that

– b lies in N (h) where h = s · J_{2n+1},

– u_{+} and v_{+} are BC-matching,

– u_{∗} can be identified with v_{∗} under an isomorphism Cent(b_{∗}, Sp_{2g}(O_{F})) ≃ Aut(h_{∗})

So let us assume that the right hand side of (v) does not vanish. Then there exists η′ ∈ Sp_{2n}(F) stably conjugate to η which has a nonvanishing orbital integral, i.e. can be conjugated into Sp_{2n}(O_{F}). We can assume that η′ ∈ Sp_{2n}(O_{F}) and that its topological Jordan decomposition satisfies η′ = b′ · v′ = v′ · b with residually semisimple b′ = (1_{2(n−g)}, b_{∗}) ∈ Sp_{2(n−g)}(O_{F}) × Sp_{2g}(O_{F}) and topologically unipotent v′. We write v′ in the form v′ = ((v′_{+})^{2}, (u′_{∗})^{2}) with v′_{+} ∈ Sp_{2(n−g)}(O_{F}) and u′_{∗} ∈ Cent(b_{∗}, Sp_{2g}(O_{F})) using 3.6(4) and the general assumption p ≠ 2. Thus we have nonvanishing O_{h′}(1, Sp_{2n}) and get from the Kazhdan-lemma 5.5 and lemma 5.6:

(vi)  O_{η′}(1, Sp_{2n}) = O_{v′}(1, Cent(b′, Sp_{2n}))
= O_{(v′_{+})^{2}}(1, Sp_{2(n−g)}) · O_{(u′_{∗})^{2}}(1, Cent(b_{∗}, Sp_{2g}))
= O_{v′_{+}}(1, Sp_{2(n−g)}) · O_{u′_{∗}}(1, Cent(b_{∗}, Sp_{2g})).

Hence the stable orbital integral O_{v′_{+}}^{st}(1, Sp_{2(n−g)}) (being the sum of integrals of nonnegative functions) is strictly positive.

By remark 5.4 there exists a BC-matching between v′_{+} and some u′_{+} ∈SO_{2(n−g)+1}(F). Then the equation (BC_{n−g}) implies that there exists u_{+} ∈ SO_{2(n−g)+1}(F) with strictly positive orbital integral and BC-matching with v′_{+}, i.e. we can assume u_{+} ∈ SO_{2(n−g)+1}(O_{F}).

Let h = sJ ∈ GL_{2n+1}(O_{F})_{RΘss} be a residually semisimple element with N (h) = b′ and define the element γ′Θ = (u_{+}, u′_{∗}) · sΘ = sΘ · (u_{+}, u′_{∗}) ∈ \begin{matrix} \\ GL & 2n+1 \end{matrix}(O_{F}). Here we identify the Cent(sΘ, GL_{2n+1} ≃ G^{h,Θ} ≃ O(q_{+}, R) × Cent (C(q_{∗}), Sp(p_{∗})) ≃ SO_{2(n−g)+1} × Cent(b_{∗}, Sp_{2g}, so that (u_{+}, u′_{∗}) can be viewed as an element of the left hand side. The element \overline{γ′}Θ ∈ \begin{matrix} \\ PGL & 2n+1 \end{matrix}(O_{F}) matches with η′ (and therefore also with η) by lemma 6.6 and therefore lies in the stable conjugacy class of γΘ.

If the left hand side of (v) does not vanish, it is immediate that there exists γ′Θ ∈ \begin{matrix} \\ GL & 2n+1 \end{matrix}(O_{F}) in the stable conjugacy class of γΘ. By reversing the above arguments we see that there exists η′ ∈ Sp_{2n}(O_{F}) in the stable class of η. So excluding the tautological case that (v) means 0 = 0 we may assume without loss of generality that γ ∈ GL_{2n+1}(O_{F}) and η ∈ Sp_{2n}(O_{F}). We may furthermore assume that γΘ = (u_{+}, u_{∗}) · sΘ and η = (v_{+}^{2}, u_{∗}^{2}) · b are the topological Jordan decompositions with BC-matching u_{+} and v_{+} and matching residually semisimple sΘ and b.

\underline{Step 2} (Calculation of the symplectic orbital integral): If η′ ∈ Sp_{2n}(F) is stable conjugate to η then the residually semisimple parts b′ and b are stable conjugate as well. If η′ has nonvanishing orbital integral then η′ and therefore also b′ can be conjugated into Sp_{2n}(O_{F}), i.e. we can assume b′ ∈ Sp_{2n}(O_{F}). By the Kottwitz lemma 4.10 b′ and b are conjugate over Sp_{2n}(O_{F}) i.e. we can assume b′ = b. This means that we obtain all relevant conjugacy classes in the stable conjugacy class of η if we let v′_{+} vary through a set of representatives for the conjugacy classes inside the stable conjugacy class of v_{+} in Sp_{2(n−g)}(F) and u′_{∗} through a set of representatives for the conjugacy classes inside the stable conjugacy class of u_{∗} in Cent(b_{∗}, Sp_{2g}). Then the corresponding η′ are of the form

η′ = b · ((v′_{+})^{2}, (u′_{∗})^{2}).

We get using (vi) and lemma 5.6:

(vii) O_{η}^{st}(1, Sp_{2n}) = ∑_{v′_{+}∼v_{+}} O_{(v′_{+})^{2}}(1, Sp_{2(n−g)}) · ∑_{u′_{∗}∼u_{∗}} O_{(u′_{∗})^{2}}(1, Cent(b_{∗}, Sp_{2g}))

= ∑_{v′_{+}∼v_{+}} O_{v′_{+}}(1, Sp_{2(n−g)}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(b_{∗}, Sp_{2g})).

\underline{Step 3} (Calculation of the Θ-twisted orbital integral): We can repeat this argument in the Θ-twisted situation, since by lemma 4.11(b) the class of the residually semisimple part s̄Θ of γ̄Θ is the only PGL_{2n+1}(F)-conjugacy class inside the stable class of s̄Θ, which meets PGL_{2n+1}(O_{F}). If we denote by u′_{+} a set of representatives for the SO_{2(n−g)+1}(F)-conjugacy classes in the stable class of u_{+} ∈ SO_{2(n−g)+1}(O_{F}) we therefore get using proposition 6.5

(viii) O_{γ̄Θ}^{st}(1, { \atop PGL}_{2n+1}) = ∑_{(u′_{+},u′_{∗})∼(u_{+},u_{∗})} O_{(u′_{+},u′_{∗})}(1, SL_{2n+1}^{sΘ})
= ∑_{u′_{+}∼u_{+}} O_{u′_{+}}(1, SO_{2(n−g)+1}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(b_{∗}, Sp_{2g})).

\underline{Step 4} (End of the proof): Since v_{+} and u_{+} are BC-matching it only remains to apply (BC_{n−g}) in order to identify

∑_{v′_{+}∼v_{+}} O_{v′_{+}}(1, Sp_{2(n−g)}) with ∑_{u′_{+}∼u_{+}} O_{u′_{+}}(1, SO_{2(n−g)+1}).

34 6. Comparison between PGL_{2n+1} and Sp_{2n} Thus the right hand sides of (vii) and (viii) coincide, and the proof of the theorem is finished.

7 Comparison between GL_{2n} × GL_{1} and GSpin_{2n+1}

Lemma 7.1 (Cayley transformation again). For a symmetric matrix q ∈ GL_{n}(R) the following holds:

(a) We have a bijection

C̃ : Alt_{n}(R)_{q−ess} → O(q, R)_{ess}, p ↦ (p − q)^{−1} · (q + p) = −N_{l}(p + q)

between the set Alt_{n}(R)_{q−ess} of skew-symmetric matrices p such that p ± q ∈ GL_{n}(R) and the set O(q, R)_{ess} of orthogonal transformations b such that b−1 ∈ GL_{n}(R). The inverse map is C̃^{−1} : b ↦ q · (b + 1) · (b − 1)^{−1}.

(b) C̃ induces a bijection between those elements q of Alt_{n}(R)_{q−ess}, for which p+q is R-Θ-semisimple, and the R-semisimple elements of O(q, R)_{ess}.

(c) The map C̃ satisfies C̃(^{t}g · p · g) = g^{−1} · C̃(p) · g for g ∈ O(q, R).

(d) We have det(b) = (−1)^{n} for b ∈ O(q, R)_{ess}.

Proof: (a) For p ∈ Alt_{n}(R)_{q−ess} we put h = p+q and b = (p−q)^{−1}·(q+p) = −^{t}h^{−1}·h Adding the formulas (iii) and (iv) in the proof of lemma 4.7 we get (−^{t}b)·q·(−b) = q, i.e. b ∈ O(q, R). Furthermore b − 1 = (p − q)^{−1} · ((p + q) − (p − q)) = (p − q)^{−1} · 2q ∈ GL_{n}(R) by the assumptions. The map C̃ is therefore defined. Conversely we get for b ∈ O(q, R)_{ess} and p = q · (b + 1) · (b − 1)^{−1} the equivalences:

p = −^{t}p ⇔ q · (b + 1) · (b − 1)^{−1} = ^{t}(b − 1)^{−1} ·^{t}(b + 1) · (−q)
⇔ (^{t}b − 1)q(b + 1) = (^{t}b + 1)q(1 − b)
⇔ ^{t}bqb +^{t}bq − qb − q = −^{t}bqb +^{t}bq − qb + q
⇔ ^{t}bqb = q ⇔ b ∈ O(q, R).

Furthermore p±q = q·((b + 1) ± (b − 1))·(b−1)^{−1} ∈ GL_{n}(R) since (b−1)^{−1}, 2b, 2, q ∈ GL_{n}(R). Therefore the map C̃^{−1} is also well defined. An easy calculation using the relation (b + 1) · (b − 1)^{−1} = (b − 1)^{−1} · (b + 1) shows that the maps C̃ and C̃^{−1} are inverse to another in their domain of definition.

(b) and (c) follow as in the proof of lemma 4.7.

(d) is clear since every b ∈ O_{n}(R) with det(b) = (−1)^{n−1} has 1 as an eigenvalue. (Alternatively we can use (a) and the computation det(−^{t}h^{−1} · h) = (−1)^{n}.)

Lemma 7.2. If q is a unimodular symmetric bilinear form on a free R-module N and b ∈ O(q, R) is R-semisimple then there exists a b-invariant q-orthogonal direct sum decomposition N = N_{1} ⊕ N_{∗} such that b acts as identity on N_{1} and b|N_{∗} ∈ O(q_{∗}, R)_{ess}, where q_{∗} is the restriction of q to N_{∗}.

36 7. Comparison between GL_{2n} × GL_{1} and GSpin_{2n+1}

Proof: The proof of lemma 4.8 can be adapted with obvious modifications.

(7.3) The explicit norm map N. Let (GL_{2n}(R) × R^{×})_{RΘss}/traf be the set of transformation classes of R-Θ-semisimple elements (h, a) ∈ GL_{2n}(R) × R^{×} under the transformations (h, a) ↦ (^{t}ghg, det g^{−1} · a) for g ∈ GL_{2n}(R), a ∈ R^{×}. Similarly let SO_{2n+1}(R)_{Rss}/conj be the set of conjugacy classes of R-semisimple elements in SO_{2n+1}(R). We define a norm map

N : (GL_{2n}(R) × R^{×})_{RΘss}/traf ⟶ SO_{2n+1}(R)_{Rss}/conj

as follows: If (h, a) ∈ GL_{2n}(R) × R^{×} represents a class of the left hand side and if h = p+q is the decomposition in the symmetric part q and the skew-symmetric part p, we decompose M = R^{2n} = M_{+} ⊕M_{−} ⊕M_{0} as in lemma 4.6. The form q_{∗} = q_{+} ⊕q_{0} on M_{∗} = M_{+} ⊕ M_{0} is unimodular. Since the ranks of M and M_{−} are even we have M_{∗} ≃ R^{2r} for some r ∈ ℕ_{0}. Let p′_{∗} and q′_{∗} be the 2r × 2r-matrices which describe p_{∗} and q_{∗} with respect to the standard basis ob R^{2r}. Let q̃_{−} be a symmetric bilinear form on M̃_{−} := R^{2(n−r)+1} such that Δ(q′_{∗}) · Δ(q̃_{−}) ∈ (R^{×})^{2}. By lemma 4.4 we have an isomorphism of quadratic spaces

i : (M_{∗}, q_{∗}) ⊕ (M̃_{−}, q̃_{−}) ˜ ⟶ (R^{2n+1}, J_{2n+1})

(observe det(J_{2n+1}) = 1) which induces an injection

j : O(M_{∗}, q_{∗}) × O(M̃_{−}, q̃_{−}) ↪ O (M_{∗} ⊕ M̃_{−}, q_{∗} ⊕ q̃_{−}) ˜→ O_{2n+1}.

This injection is canonical (i.e. independent of the chosen isomorphism i) on the set of conjugacy classes. Now N (h), the image of the class of h under N, is defined to be the O_{2n+1}(R)- conjugacy class of j(C̃(p′_{∗}), 1_{2(n−r)+1}) ∈ O_{2n+1}(R), where we use the Cayley-trans-form-map C̃ with respect to q′_{∗} from lemma 7.1. We observe that det(C̃(p′_{∗})) = 1 by lemma 7.1(d) and therefore N (h) lies in SO_{2n+1}(R). Since the centralizer of j(C̃(p′_{∗}), 1_{2(n−r)+1}) in O_{2n+1}(R) contains {1_{2r}} × O_{2(n−r)+1}(R) i.e. elements of determinant −1, the O_{2n+1}(R)-conjugacy class is in fact a SO_{2n+1}(R)-conjugacy class.

Lemma 7.4. In the notations of 7.3 the spinor norm of N (h) is the class of det(h) mod (R^{×})^{2}.

Proof: It is sufficient to consider the case R = F, since we have an injection O_{F}^{×}/(O_{F}^{×})^{2} ↪ F^{×}/(F^{×})^{2}. If σ denotes the spinor norm of N (h) we have by a theorem of Zassenhaus (comp. [Zas]) in the version of [Mas]

σ ≡ det (id − C̃(p′_{∗})) · Δ(q′_{∗}) mod (R^{×})^{2}.

But id−C̃(p′_{∗}) = (q′_{∗}−p′_{∗})^{−1}·2·q′_{∗} so that we get σ ≡ det(q′_{∗}−p′_{∗})^{−1}·2^{2r} ≡ det(q′_{∗}−p′_{∗}) mod (R^{×})^{2}. Furthermore det(q′_{∗} − p′_{∗}) = det^{t}(q′_{∗} − p′_{∗}) = det(p′_{∗} + q′_{∗}). Since the discriminant of p_{−} is a square we finally get σ ≡ det(p′_{∗} + q′_{∗}) · det(p_{−}) ≡ det(h) mod (R^{×})^{2}.

Proposition 7.5.

(a) The map N : (GL_{2n}(R)×R^{×})_{RΘss}/traf ⟶ SO_{2n+1}(R)_{Rss}/conj is well defined and surjective. Two classes lie in the same fiber iff they have representatives of the form (h, a_{1}) and (h, a_{2}).

(b) If (h, a) represents a class in (GL_{2n}(R) × R^{×})_{RΘss}/traf then (h · J^{−1}, a)Θ ∈ (GL_{2n}(R) × R^{×}) ⋊ ⟨Θ⟩ matches in the sense of Θ-endoscopy with some element η ∈ GSpin_{2n+1}(R), which maps to N (h) under the projection pr_{ad} : GSpin_{2n+1} → SO_{2n+1}.

Proof: (a) If we replace p′_{∗} by some ^{t}g · p′_{∗} · g for g ∈ O(q′_{∗}, R), this does not change the conjugacy class of N (h) by lemma 7.1(c). Since the effect of the other choices has already been considered, the map N is well defined. To prove surjectivity first observe that each class b ∈ SO_{2n+1}(R)_{Rss}/conj can be represented after some transformation of J_{2n+1} in the form (b′, 1_{2(n−r)+1}) with b′ ∈ SO(q′_{∗}, R)_{ess} by lemma 7.2 with a unique r ≤ n and some symmetric q′_{∗} ∈ GL_{2r}(R). One should think of (b′, 1_{2(n−r)+1}) as a block-matrix

(\begin{matrix}B_{11} & 0 & B_{12} \\ 0 & 1_{2(n−r)+1} & 0 \\ B_{21} & 0 & B_{22} \end{matrix}) with b′ = (\begin{matrix}B_{11} & B_{22} \\ B_{21} & B_{22} \end{matrix})

Since the class of Δ(q′_{∗}) in R^{×}/(R^{×})^{2} is the inverse of the class of Δ(J_{2n+1}| ker(b−1)), the transformation class of q′_{∗} is unique by lemma 4.4. Up to this the SO(q′_{∗}, R)- conjugacy-class of b′ is unique. The bijectivity of the Cayley-transform map and property 7.1(c) then imply that there is a p′ ∈ Alt_{2r}(R), which is unique up to transformations with elements of SO(q′_{∗}, R), such that b = C̃(p′). Now we consider the unimodular bilinear form h′ = p′ + q′ on R^{2r}, which is unique up to transformations with elements of GL_{2r}(R), and some unimodular skew symmetric form p_{−} on R^{2(n−r)}. The form p_{−} ⊕ h′ on R^{2n} is then unimodular and R-Θ-semisimple i.e. corresponds to a R-Θ-semisimple transformation class h. For every R-semisimple a ∈ R^{×} we get N (h, a) = b. Since the transformation class of h′ is unique by the above considerations and by lemma 4.3 we conclude that the fibers of N correspond to the different choices for the R-semisimple element a ∈ R^{×}.

(b) At first we consider the case that R = F̄ is an algebraically closed field, so that we may assume that γ = h · J_{2n+1}^{−1} has diagonal form γ = diag(t_{1}, …, t_{2n}). After applying a permutation in W_{Sp_{2n}} we may assume

(i) t_{i} ≠ t_{2n+1−i} for i ≤ r and t_{i} = t_{2n+1−i} for r + 1 ≤ i ≤ 2n − r.

We have:

h = antidiag(t_{1}, −t_{2}, t_{3}, …, −t_{2n})
h ±^{t}h = antidiag(t_{1} ∓ t_{2n}, −(t_{2} ∓ t_{2n−1}), t_{3} ∓ t_{2n−2}, …, −(t_{2n} ∓ t_{1}))
−^{t}h^{−1} · h = diag(t_{2n}/t_{1}, t_{2n−1}/t_{2}, …, t_{1}/t_{2n})

Thus M_{−} ≃ R^{2(n−r)} is generated by the standard basis elements e_{r+1}, …, e_{2n−r} of R^{2n} and M_{∗} = M_{−} ⊕ M_{0} has the basis e_{1}, e_{2}, , …, e_{r}, e_{2n+1−r}, …, e_{2n}. The matrix of the symmetric bilinear form given by ^{t}d · (h +^{t}h) · d with respect to this latter basis has the standard form J′_{2r}, if we take d = diag((t_{1} − t_{2n})^{−1}, (t_{2} − t_{2n−1})^{−1}, …, (t_{r} − t_{2n+1−r})^{−1}, 1, …, 1). Since d and ^{t}h^{−1} · h commute we get from the definition of N (h) that N (h) is represented by the diagonal matrix

diag(t_{2n}/t_{1}, t_{2n}/t_{2}, …, t_{n+1}/t_{n}, 1, t_{n}/t_{n+1}, …, t_{1}/t_{2n}),

which can be conjugated into the form of example 1.14. This proves the claim in the case that R is an algebraically closed field.

In the case that R is arbitrary we consider the commutative diagram with exact rows and columns and a connecting homomorphism marked with … (snake lemma):

1

↓

1  ⟶ {±1} ⟶ …
↓  ↓
1 ⟶ Spin_{2n+1}(R)  Spin_{2n+1}(R) ⟶ 1
↓  ↓  ↓
1 ⟶ R^{×} ⟶ GSpin_{2n+1}(R) ⟶^{pr_{ad}} SO_{2n+1}(R) ⟶ 1
↓  ↓^{μ}  ↓^{Spinnorm}
… ⟶ R^{×} ⟶^{r↦r^{2}}  R^{×}  ⟶ R^{×}/(R^{×})^{2} ⟶ 1
↓  ↓  ↓
1  1  1

It follows from this diagram and lemma 7.4 that a matrix η_{0} in the class N (h) has a preimage η ∈ GSpin_{2n+1}(R) such that μ(η) = det(h) · a^{2} and that the set {x ∈ GSpin_{2n+1}(F̄)|pr_{ad}(x) = η_{0}, μ(x) = det(h) · a^{2}} just consists of ±η. On the other hand by example 1.15 an element η′ ∈ GSpin_{2n+1}(F̄) matching with (h, a) satisfies μ(η′) = det(h) · a^{2}. From the validity of the proposition over F̄ now follows that either η or −η matches with (h, a). This element has all desired properties.

Remark: To get (b) it is even in the case R = F not sufficient just to apply Steinbergs theorem on rational elements to the rational conjugacy class inside GSpin_{2n+1}(F̄), which matches with (h · J^{−1}, a)Θ: If η′ ∈ GSpin_{2n+1}(F) denotes such an element, then we only know from the case R = F̄ that pr_{ad}(η′) and N (h) are stably conjugate elements in SO_{2n+1}(F). But the Spinor norm is not invariant under stable conjugation. Thus it is not clear without the use of lemma 7.4 that N (h) can be lifted to a class in GSpin_{2n+1}(F), on which the multiplier μ takes the correct value.

Corollary 7.6. For each semisimple η ∈ GSpin_{2n+1}(F) there exists an F -Θ-semisimple (h · J^{−1}, a)Θ ∈ (GL_{2n}(F) × F^{×}) ⋊ ⟨Θ⟩ matching with η.

Proof: By 7.5(a) for R = F there exists (h, a_{1}) ∈ GL_{2n}(F)×F^{×} with pr_{ad}(η) ∈ N (h) and by (b) there exists η_{1} ∈ GSpin_{2n+1}(F) matching with (h · J^{−1}, a_{1})Θ such that pr_{ad}(η_{1}) = pr_{ad}(η). It follows η = η_{1} · b for some b ∈ F^{×} ≃ Center(GSpin_{2n+1}(F)). Then (h · J^{−1}, a_{1} · b)Θ matches with η.

Lemma 7.7. For G = GL_{2n} × 𝔾_{m} let γ_{1}, γ_{2} ∈ G(O_{F}), g_{F} ∈ G(F̄) be such that γ_{2}Θ = g_{F} · γ_{1}Θ · g_{F}^{−1} with Θ as in example 1.9. Then there exists g_{R} ∈ G(O_{F}) with γ_{2}Θ = g_{R} · γ_{1}Θ · g_{R}^{−1}.

Proof: Write γ_{i} = (h_{i} · J_{2n}^{−1}, a_{i}), g_{F} = (h_{F} , ã). Then the assumption means: h_{2} = h_{F} · h_{1} · ^{t}h_{F} and a_{2} = ã · a_{1} · det(h_{F})^{−1} · ã^{−1} which implies det(h_{F}) ∈ O_{F}^{×}. By lemma 4.11(a) there exists h_{R} ∈ GL_{2n}(O_{F}) with h_{2} = h_{R} · h_{1} · ^{t}h_{R}. This implies det(h_{R})^{2} = det(h_{F})^{2}. If det(h_{R}) = − det(h_{F}) then h_{F}^{−1} · h_{R} ∈ O(h_{1})(F̄) where O(h_{1}) = {h ∈ GL_{2n} | ^{t}h·h_{1}·h = h_{1}} = O(q_{+,1})×(Sp(p_{−,1} ⊕ p_{0,1}) ∩ O(q_{−,1} ⊕ q_{0,1})) has determinant −1. This implies M_{+,1} ≠ 0 so that we get an element h_{ϵ} ∈ O(h_{1})(O_{F}) of determinant −1. Replacing h_{R} by h_{R} · h_{ϵ} we can now assume det(h_{R}) = det(h_{F}). With g_{R} = (h_{R}, 1) we now have γ_{2}Θ = g_{R} · γ_{1}Θ · g_{R}^{−1}.

Lemma 7.8. Let (h, a) = (sJ, a) ∈ GL_{2n}(O_{F}) × O_{F}^{×} be O_{F} -Θ-semisimple and b = (1_{2(n−r)+1}, b_{∗}) ∈ SO_{2n+1}(O_{F}) a representing element of N (h). With p_{∗}, q_{∗}, p_{−} as in 7.3 assume that we have matching topologically unipotent elements u_{−} ∈ Sp_{2(n−r)}(F) ≃ Sp(p_{−}) and v_{−} ∈ SO_{2(n−r)+1}(F) ≃ (ker(b − 1)(F) ∩ SO_{2n+1}(F)) and an additional topologically unipotent element u_{∗} ∈ SO(q_{∗})(F) ∩ Sp(p_{∗})(F) ≃ Cent(b_{∗}, SO(q_{∗})(F)). Then the element γΘ = (s, a)Θ·(u_{−}, u_{∗}) = (u_{−}, u_{∗})·(s, a)Θ ∈ (GL_{2n}(F) × F^{×})⋊⟨Θ⟩ matches with some element η ∈ GSpin_{2n+1}(F), which projects to β := (v_{−}^{2}, u_{∗}^{2}) · b = b · (v_{−}^{2}, u_{∗}^{2}) ∈ SO_{2n+1}(F).

Proof: We first prove the existence of η ∈ GSpin_{2n+1}(F̄) with the desired properties and thus work over the algebraically closed field F = F̄ as in the proof of lemma 6.3(c). Thus we may assume that γ and β lie in the corresponding diagonal tori. The same holds for the residually semisimple parts s resp. b and the topologically unipotent parts u = (u_{+}, u_{∗}) and v = (v_{−}^{2}, u_{∗}^{2}). As (s, a)Θ matches with some η_{s} which projects to b by 7.5(b) we only have to examine the topologically unipotent elements. We can make the assumption (i) and write

u_{−} = diag(w_{r+1}, …, w_{n}, w_{n}^{−1}, …, w_{r+1}^{−1}) ∈ Sp_{2(n−r)}(F̄)
u_{∗} = diag(w_{1}, …, w_{r}, w_{r}^{−1}, …, w_{1}^{−1}) ∈ SO_{2r}(F̄)^{b_{∗}}.

By the definition of BC-matching we can assume

v_{−} = diag(w_{r+1}, …, w_{n}, 1, w_{n}^{−1}, …, w_{r+1}^{−1}) ∈ SO_{2(n−r)+1}(F̄).

Taking everything together we get by the description of M_{−} and M_{∗} in the proof of lemma 7.5(b):

u = (w_{1}, …, w_{n}, w_{n}^{−1}, …, w_{1}^{−1})
v = (w_{1}^{2}, …, w_{n}^{2}, 1, w_{n}^{−2}, …, w_{1}^{−2})

and the matching between γΘ and some η ∈ GSpin_{2n+1}(F̄) which projects to β follows from example 1.14.

To get η as an element of GSpin_{2n+1}(F) we observe that the determinant of γJ_{2n}^{−1} equals the spinor norm of β as an element of F^{×}/(F^{×})^{2}: This is already clear by 7.4 for the residually semisimple parts, but both topologically unipotent parts lead to the neutral element in F^{×}/(F^{×})^{2}, since 2 ≠ p by assumption. Now one argues as in the proof of 7.5(b) to get η as an F -rational element.

Theorem 7.9. (BC_{2}) is true.

Proof: We observe that every pair of BC-matching (topologically unipotent) elements γ̄ ∈ SO_{5}(F) and η_{1} ∈ Sp_{4}(F) can be obtained from a pair of (topologically unipotent) elements γ ∈ GSp_{4}(F) ≃ GSpin_{5}(F) and ηΘ = Θη ∈ (G̃L_{4} × GL_{1})(F) such that γ̄ = pr_{ad}(γ) and η = (η_{1}, a) ∈ (GL_{4} × GL_{1})^{Θ}(F) ≃ (Sp_{4} × GL_{1})(F) and such that γ^{2} matches with ηΘ in the sense of 1.11. This follows immediately from the definition of BC-matching 1.12 and example 1.14.

If we apply lemma 5.8 in the case G = GSpin_{5} ≃ GSp_{4}, T = 𝔾_{m}, H = SO_{5} and lemma 5.6 we get

O_{γ^{2}}^{st}(1, GSp_{4}) = Ost ; γ̄(1, SO_{5})

Since we have O_{η}^{st}(1, Sp_{4} × GL_{1}) = O_{η_{1}}^{st}(1, Sp_{4}) by lemma 5.8 the statement of (BC_{2}) is equivalent to the identity

O_{γ}^{st}(1, GSp_{4}) = O_{η}^{st}(1, Sp_{4} × GL_{1})

for matching topologically unipotent γ ∈ GSp_{4}(F) and ηΘ = Θη ∈ (G̃L_{4}×GL_{1})(F). In the case that η is strongly Θ-regular this has been proved in [Fl2, ch. II]. The general case follows by the germ expansion principle as in [Hal3], [Rog].

Corollary 7.10 (Fundamental lemma for Sp_{4} ↔ { \atop PGL}_{5}). If γΘ ∈ { \atop PGL}_{5}(F) and h ∈ Sp_{4}(F) are matching semisimple elements then we have

O_{γΘ}^{st}(1, { \atop PGL}_{5}) = O_{h}^{st}(1, Sp_{4}).

Proof: This follows from theorem 7.9, (BC_{1}) (compare 5.3) and theorem 6.7.

Theorem 7.11. Let G = GL_{2n} × 𝔾_{m}. If γΘ ∈ G̃(F) and η ∈ GSpin_{2n+1}(F) are matching semisimple elements and if conjecture (BC_{m}) is true for all m ≤ n then we have

(ii) O_{γΘ}^{st}(1, G̃) = O_{η}^{st}(1, GSpin_{2n+1}).

Proof: Let h = pr_{ad}(η) ∈ SO_{2n+1}(F). In view of lemma 5.8 we have to prove

O_{γΘ}^{st}(1, G̃) = O_{h}^{st}(1, SO_{2n+1}). (iii)

The proof is now similar to the proof of Theorem 6.7.

\underline{Step 1}: Let us assume that the right hand side of (iii) does not vanish. Then there exists h′ ∈ SO_{2n+1}(F) stably conjugate to h which has a nonvanishing orbital integral, i.e. can be conjugated into SO_{2n+1}(O_{F}). We can assume that h′ ∈ SO_{2n+1}(O_{F}). Since h′ = pr_{ad}(η′) for some η′ ∈ GSpin_{2n+1}(F) in the stable conjugacy class of η we can assume without loss of generality that η′ = η and thus h′ = h. Furthermore we can assume that the topological Jordan decomposition is of the form h = b · v = v · b with residually semisimple b = (1_{2(n−r)+1}, b_{∗}) ∈ SO_{2(n−r)+1}(O_{F}) × SO(q_{∗}, O_{F})_{ess} and topologically unipotent v, where q_{∗} denotes the restriction of J_{2n+1} to the orthogonal complement of ker(b − 1) ≃ O_{F}^{2(n−r)+1}. Here we observe that the restriction of J_{2n+1} to ker(b − 1) can be assumed to be a multiple of the standard form J_{2(n−r)+1} and that b_{∗} has determinant 1 as 1_{2(n−r)+1} has determinant 1. We can write v in the form v = ((v_{−})^{2}, (u_{∗})^{2}) with v_{−} ∈ SO_{2(n−r)+1}(O_{F}) and u_{∗} ∈ Cent (b_{∗}, SO(q_{∗}, O_{F})) since 2 ∈ ℤ_{p}^{×}.

We remark that condition (∗) in the Kazhdan-lemma 5.5 is satisfied by lemma 7.7, so that we get nonvanishing

O_{h}(1, SO_{2n+1}) = O_{v}(1, Cent(b, SO_{2n+1}))

Observing that we have an isomorphism

Cent(b, SO_{2n+1}) ≃ SO_{2(n−r)+1} × Cent(b_{∗}, SO(q_{∗})) × {±1}

we can decompose the orbital integral on the right hand side using lemma 5.7:

O_{h}(1, SO_{2n+1}) = O_{(v_{−})^{2}}(1, SO_{2(n−r)+1}) · O_{(u_{∗})^{2}}(1, Cent(b_{∗}, SO(q_{∗})))
= O_{v_{−}}(1, SO_{2(n−r)+1}) · O_{u_{∗}}(1, Cent(b_{∗}, SO(q_{∗}))),

(use lemma 5.6 in the last step) i.e. O_{v_{−}}^{st}(1, SO_{2(n−r)+1}) (being the sum of integrals of nonnegative functions) is strictly positive. Since v_{−} is BC-matching with some u′_{−} ∈ Sp_{2(n−r)}(F) the equation (BC_{n−r}) implies that there exists u_{−} ∈ Sp_{2(n−r)}(F) matching with v_{−} and with strictly positive orbital integral, i.e. we can assume u_{−} ∈ Sp_{2(n−r)}(O_{F}). Let s ∈ GL_{2n}(O_{F}) be a residually semisimple element with N (s·J_{2n}^{−1}, a) = b for some a ∈ O_{F}^{×}, chosen such that we can identify the corresponding q_{∗} on M_{∗} with the above obtained q_{∗}. By modifying a we can assume that (s, a)Θ matches with the residually semisimple part η_{s} of η. We define the element γ′Θ = (u_{−}, u_{∗}) · sΘ = sΘ · (u_{−}, u_{∗}) ∈ G̃(O_{F}). The element γ′Θ ∈ G̃(O_{F}) matches with h by lemma 7.8 and therefore lies in the stable conjugacy class of γΘ.

If the left hand side of (iii) does not vanish, it is immediate that there exists γ′Θ ∈ G̃(O_{F}) in the stable conjugacy class of γ_{Θ}. By reversing the above arguments we see that there exists h′ ∈ SO_{2n+1}(O_{F}) in the stable class of h. So excluding the tautological case that (ii) means 0 = 0 we may assume without loss of generality that γ ∈ G(O_{F}) and h ∈ SO_{2n+1}(O_{F}). We may furthermore assume that γΘ = (u_{+}, u_{∗}) · sΘ and h = (v_{+}^{2}, u_{∗}^{2}) · b are the topological Jordan decompositions with BC-matching u_{+} and v_{+} and matching residually semisimple sΘ and η_{s}.

\underline{Step 2}: As in the proof of theorem 6.7 we get from lemma 4.10 the fact that we obtain all relevant conjugacy classes in the stable conjugacy class of h if we let v′_{−} vary through a set of representatives of the stable conjugacy class of v_{−} in SO_{2(n−r)+1}(F) and u′_{∗} through a set of representatives of the stable conjugacy class of u_{∗} in Cent(b_{∗}, SO(q_{∗})) and then consider all h′ = b · ((v′_{+})^{2}, (u′_{∗})^{2}). i.e.

(iv) O_{h}^{st}(1, SO_{2n+1}) = ∑_{v′_{+}∼v_{+}} O_{(v′_{+})^{2}}(1, SO_{2(n−r)+1}) ·∑_{u′_{∗}∼u_{∗}} O_{(u′_{∗})^{2}}(1, Cent(b_{∗}, SO(q_{∗})))

= ∑_{v′_{+}∼v_{+}} O_{v′_{+}}(1, SO_{2(n−r)+1}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(b_{∗}, SO(q_{∗}))).

\underline{Step 3}: We can repeat this argument in the Θ-twisted situation, since by lemma 7.7 the class of the residually semisimple part (s, a)Θ of γΘ is the only G(F)- conjugacy class inside the stable class of (s, a)Θ, which meets G(O_{F}) and since the Kazhdan-Lemma 5.5 holds for G̃ by the same lemma. We remark that G^{(s,a)Θ} ≃ Sp_{2(n−r)} × Cent(b_{∗}, SO(q_{∗})) × 𝔾_{m} by the definition of Θ and lemma 4.9(e), so G^{(s,a)Θ} is connected and we can use lemma 5.8 to get rid of the 𝔾_{m} factors in the following orbital integrals. If we denote by u′_{−} a set of representatives for the Sp_{2(n−r)}(F)- conjugacy classes in the stable class of u_{−} ∈ Sp_{2(n−r)}(O_{F}) we get

(v) O_{γ̄Θ}^{st}(1, G̃) = ∑_{(u′_{−},u′_{∗})∼(u_{−},u_{∗})} O_{(u′_{−},u′_{∗})}(1, G^{sΘ})

= ∑_{u′_{−}+∼u_{−}} O_{u′_{−}}(1, Sp_{2(n−r)}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(b_{∗}, SO(q_{∗}))).

\underline{Step 4}: Since v_{−} and u_{−} are BC-matching we can apply (BC_{n−r}) to get that the right hand sides of (iv) and (v) coincide, which proves the theorem.

8 Comparison between SO_{2n+2} and Sp_{2n}

Let R be as in 4.2.

Lemma 8.1. Let N be a free R-module.

(a) If p is a unimodular symplectic form on N and if β ∈ Sp(p, R) is R-semisimple then there exists a β-invariant orthogonal (with respect to p) direct sum decomposition N = N_{+} ⊕ N_{−} ⊕ N_{∗} such that β acts as identity on N_{+}, as −id on N_{−} and β_{∗} = β|N_{∗} ∈ Sp(p_{∗}) satisfies β_{∗} − β_{∗}^{−1} ∈ GL(N_{∗}), where p_{∗} is the restriction of p to N_{∗}.

(b) If q is a unimodular symmetric bilinear form on N and b ∈ O(q, R) is Rsemisimple then there exists a b-invariant orthogonal (with respect to q) direct sum decomposition N = N_{+} ⊕ N_{−} ⊕ N_{∗} such that b acts as identity on N_{+}, as −id on N_{−} and b_{∗} = b|N_{∗} ∈ O(q_{∗}) satisfies b_{∗} − b_{∗}^{−1} ∈ GL(N_{∗}), where q_{∗} is the restriction of q to N_{∗}.

Proof: The proof of lemma 4.8 can be adapted with obvious modifications: We have b − b^{−1} = b^{−1} · (b − 1) · (b + 1), so that b − b^{−1} ∈ GL(N_{∗}) is equivalent to b − 1, b + 1 ∈ GL(N_{∗}).

Lemma 8.2. Let b ∈ GL_{n}(R) satisfy b − b^{−1} ∈ GL_{n}(R). Then the following holds:

(a) If q ∈ GL_{n}(R) is symmetric and b ∈ O(q, R) then the matrix p = q · (b − b^{−1}) is unimodular skew-symmetric and we have b ∈ Sp(p, R).

(b) If p ∈ GL_{n}(R) is skew-symmetric and b ∈ Sp(p, R) then the matrix q = p · (b − b^{−1})^{−1} is unimodular symmetric and we have b ∈ SO(q, R).

(c) Under the conditions of (a) and (b) we have:

Cent(b, O(q)) = Cent(b, Sp(p)) = Cent(b, SO(q)).

(d) The above statements and formulas are invariant under the substitutions b ↦ g^{−1}bg, q ↦ ^{t}gqg, p ↦ ^{t}gpg for g ∈ GL_{n}(R).

Proof: (a) We have p = q·b−(^{t}b·q·b)·b^{−1} = qb−^{t}(qb) and^{t}bpb = ^{t}b(qb)b−^{t}bqb^{−1}·b = qb−^{t}bqb·b^{−1} = qb−qb^{−1} = p. Unimodularity of p follows from q, (b−b^{−1}) ∈ GL_{n}(R).

(b) We have^{t}q = q ⇔ −^{t}(b−b^{−1})^{−1}·p = p·(b−b^{−1})^{−1} ⇔ p(b−b^{−1}) = (−^{t}b+^{t}b^{−1})p ⇔ tb^{−1} · (^{t}bpb − p) = (p − ^{t}bpb) · b^{−1} ⇐ b ∈ Sp(p, R) and ^{t}bqb = ^{t}bp(b − b^{−1})^{−1} · b = tbpb · (b − b^{−1})^{−1} = p(b − b^{−1})^{−1} = q. As an element of a symplectic group b has determinant 1.

44 8. Comparison between SO_{2n+2} and Sp_{2n}

(c) For x ∈ Cent(b, GL_{n}(R)) we have xb = bx and b^{−1}x = xb^{−1} which imply x(b − b^{−1}) = (b − b^{−1})x, so that we get ^{t}xqx = q ⇔ ^{t}xqx(b − b^{−1}) = q(b − b^{−1}) ⇔ txq(b − b^{−1})x = q(b − b^{−1}) ⇔ ^{t}xpx = p. This proves the first ”=”. The second follows immediately since elements of Sp(p) have determinant 1.

(d) follows by almost trivial computations.

(8.3) If s ∈ O_{2n+2} with det(s) = −1 denotes a reflection, we can identify the semidirect product SO_{2n+2}⋊⟨Θ⟩ where Θ = int(s) with the orthogonal group O_{2n+2}.

Let O_{2n+2}(R)_{Rss}^{−}/conj be the set of SO_{2n+2}(R)-conjugacy classes of R-semisimple (=R-Θ-semisimple) elements of h ∈ O_{2n+2}(R) with det(h) = −1. Recall that Sp_{2n}(R)_{Rss}/conj is the set of conjugacy classes of R-semisimple elements in Sp_{2n}(R). We define a norm map

N : O_{2n+2}(R)_{Rss}^{−}/conj ⟶ Sp_{2n}(R)_{Rss}/conj

as follows: If b ∈ O_{2n+2}(R) represents a class of the left hand side, we decompose N = R^{2n+2} = N_{+} ⊕ N_{−} ⊕ N_{∗} as in lemma 8.1(b). Let b_{+} = id_{N_{+}}, b_{−} = −id_{N_{−}} and b_{∗} = b|N_{∗}. Let q_{∗} be the restriction of the form J_{2n+2} to N_{∗}. We may think of q_{∗} as a symmetric matrix after introducing a basis of N_{∗}. Since b_{∗} ∈ Sp(p_{∗}) for p_{∗} = q_{∗} · (b_{∗} − b_{∗}^{−1}) by lemma 8.2(a) we have det(b_{∗}) = 1. Therefore −1 = det(b) = det(b_{+}) · det(b_{−}) · det(b_{∗}) = 1 · (−1)^{rankN_{−}} · 1, i.e. rank(N_{−}) is odd = 1 + 2r_{−}. Since rank(N_{∗}) is even by lemma 8.2 we have rank(N_{+}) = 1 + 2r_{+} for some r_{+} ∈ ℕ_{0}. Now we equip the R-module M = M_{+} ⊕ M_{−} ⊕ N_{∗} where M_{+} ≃ R^{2r_{+}}, M_{−} ≃ R^{2r_{−}} with the alternating form p = J_{2r_{+}} ⊕ J_{2r_{−}} ⊕ p_{∗} and the linear automorphism β = id_{M_{+}} × −id_{M_{−}} × b_{∗} ∈ Sp(p). Identifying the symplectic space (M, p) ≃ (R^{2n}, J_{2n}) we can think of β as an element of Sp_{2n}(R). The conjugacy class of β in Sp_{2n}(R) does not depend on the choices we made (apply lemma 8.2(d) ) and is the desired N (b). It is clear that N (b) is R-semisimple.

Proposition 8.4. Let R be as in 4.2.

(a) The map N : O_{2n+2}(R)_{Rss}^{−}/conj ⟶ Sp_{2n}(R)_{Rss}/conj is well defined. Each b ∈ O_{2n+2}(R)_{Rss}^{−}/conj matches with N (b) in the sense of Θ-endoscopy (compare examples 1.10, 1.16).

(b) The map N is surjective, if R = O_{F}. Its fibers are of order 2 = #(R^{×}/(R^{×})^{2}) and describe the two different pairs (q_{+}, q_{−}) of classes of unimodular quadratic forms on (M_{+}, M_{−}) such that Δ(q_{+}) · Δ(q_{−}) ≡ det(q_{∗})^{−1} mod (R^{×})^{2}.

Proof: (a)That N is well defined is already clear. By the definition of matching we can work over R = F̄, so that we may assume that γ = b · s^{−1} ∈ SO_{2n+2}(R) has diagonal form γ = diag(t_{1}, …, t_{n+1}, t_{n+1}^{−1}, …, t_{1}^{−1}), where s is the reflection defined in 1.10. We have:

b = (\begin{matrix}diag(t_{1}, …, t_{n}) \\ & 0 & t_{n+1} \\ & t_{n+1}^{−1} & 0 \\ & & & diag(t_{n}^{−1}, …, t_{1}^{−1}) \end{matrix})

b − b^{−1} = diag(t_{1} − t_{1}^{−1}, …, t_{n} − t_{n}^{−1}, 0, 0, t_{n}^{−1} − t_{n}, …, t_{1}^{−1} − t_{1})

With the standard basis (e_{i})_{1≤i≤2n+2} of R^{2n+2} we get:

M_{+} = ⟨e_{i}, e_{2n+3−i} | t_{i} = 1, 1 ≤ i ≤ n⟩ ⊕ ⟨t_{n+1} · e_{n+1} + e_{n+2}⟩
M_{−} = ⟨e_{i}, e_{2n+3−i} | t_{i} = −1, 1 ≤ i ≤ n⟩ ⊕ ⟨t_{n+1} · e_{n+1} − e_{n+2}⟩
M_{∗} = ⟨e_{i}, e_{2n+3−i} | t_{i} ≠ ±1, 1 ≤ i ≤ n⟩

The corresponding description of N = N_{+} ⊕ N_{−} ⊕ N_{∗} can be arranged such that:

N_{+} = ⟨e′_{i}, e′_{2n+3−i} | t_{i} = 1, 1 ≤ i ≤ n⟩
N_{−} = ⟨e′_{i}, e′_{2n+3−i} | t_{i} = −1, 1 ≤ i ≤ n⟩
M_{∗} = ⟨e′_{i}, e′_{2n+3−i} | t_{i} ≠ ±1, 1 ≤ i ≤ n⟩

where e′_{i} = (−1)^{i} · (t_{i} − t_{i}^{−1})^{−1}e_{i} if t_{i} ≠ ±1 and 1 ≤ i ≤ n and e′_{j} = e_{j} else. With respect to this new basis of M_{∗} the symplectic form given by p_{∗} = q_{∗} · (b_{∗} − b_{∗}^{−1}) has standard form J_{2g}, so that the symplectic form p on R^{2n} can be assumed to be of standard form J_{2n} with respect to the basis e′_{1}, …, e′_{n}, e′_{n+3}, …, e′_{2n+2}. The symplectic transformation β = id_{N_{+}} × (−id_{N_{−}}) × b_{∗} in N (b) has the diagonal form diag(t_{1}, …, t_{n}, t_{n}^{−1}, …, t_{1}^{−1}) with respect to this basis. The claim now follows from example 1.16.

(b) Let β ∈ Sp_{2n}(R)_{Rss}. We decompose N = R^{2n} = N_{+} ⊕ N_{−} ⊕ N_{∗} as in lemma 8.1(a). Since this decomposition is J_{2n}-orthogonal the restrictions p_{+}, p_{−}, p_{∗} of the symplectic form J_{2n} to N_{+}, N_{−} and N_{∗} are unimodular, so these spaces have even rank: N_{+} ≃ R^{2r_{+}}, N_{−} ≃ R^{2r_{−}}, N_{∗} ≃ R^{2g}. If we view p_{∗} as skew symmetric matrix and β_{∗} ∈ Sp(p_{∗}) ⊂ SL_{2g} we can form the symmetric matrix (bilinear form) q_{∗} = p_{∗}·(β_{∗}−β_{∗}^{−1})^{−1} and get β_{∗} ∈ SO(q_{∗}). For ϵ_{±} ∈ R^{×}/(R^{×})^{2} we consider the symmetric bilinear forms q_{+} = ϵ_{+}·J_{1+2r_{+}} on M_{+} = R^{1+2r_{+}} and q_{−} = ϵ_{−}·J_{1+2r_{−}} on M_{−} = R^{1+2r_{−}}. By lemma 4.4 there are two different choices of pairs (ϵ_{+}, ϵ_{−}) such that the quadratic space (M, q) = (M_{+}, q_{+}) ⊕ (M_{−}, q_{−}) ⊕ (N_{∗}, q_{∗}) is isomorphic to the standard space (R^{2n+2}, J′_{2n+2}). For these two choices the element b = id_{M_{+}} × (−id_{M_{−}}) × β_{∗} ∈ O(q, R)^{−} can be viewed as an element of O_{2n+2}(R)_{Rss}^{−}, which maps to β under N. It is clear from the constructions that the two classes just obtained are all SO_{2n+2}(R)- conjugacy classes in O_{2n+2}(R)_{Rss}^{−} mapping to β under N.

Lemma 8.5. Let γ_{1} ∈ O_{2n+2}(O_{F})^{−} be R-Θ-semisimple.

(a) If γ_{2} := g_{F}^{−1} · γ_{1} · g_{F} ∈ O_{2n+2}(O_{F})^{−} for g_{F} ∈ SO_{2n+2}(F) there exists g_{R} ∈ SO_{2n+2}(O_{F}) with γ_{2} = g_{R}^{−1} · γ_{1} · g_{R}.

(b) There is a unique SO_{2n+2}(O_{F})-conjugacy class {γ′_{1}} in SO_{2n+2}(O_{F}) different from {γ_{1}} such that for every g_{F} ∈ SO_{2n+2}(F̄) with γ_{2} := g_{F}^{−1} · γ_{1} · g_{F} ∈ O_{2n+2}(O_{F})^{−} there either exists g_{R} ∈ SO_{2n+2}(O_{F}) with γ_{2} = g_{R}^{−1} · γ_{1} · g_{R} or g′_{R} ∈ SO_{2n+2}(O_{F}) with γ_{2} = (g′_{R})^{−1} · γ′_{1} · g′_{R}

Proof: (a) Let M = O_{F}^{2n+2} = M_{+,i} ⊕ M_{−,i} ⊕ M_{∗,i} for i = 1, 2 be the orthogonal decompositions with respect to γ_{i} as in lemma 8.1(b) and let q_{±,i}, q_{∗,i} denote the restrictions of the standard form J′_{2n+2} to these subspaces. Since M_{±,i} are eigenspaces of γ_{i} we have

g_{F} (M_{±,2} ⊗_{O_{F}} F) = M_{±,1} ⊗_{O_{F}} F

Since g_{F} ∈ SO_{2n+2}(F) we get for the orthogonal complement:

g_{F} (M_{∗,2} ⊗_{O_{F}} F) = M_{∗,1} ⊗_{O_{F}} F.

In fact g_{F} induces isomorphisms of quadratic spaces:

(M_{±,2} ⊗_{O_{F}} F, q_{±,2}) ˜→ (M_{±,1} ⊗_{O_{F}} F, q_{±,1}) and
g_{F,∗} : (M_{∗,2} ⊗_{O_{F}} F, q_{∗,2}) ˜→ (M_{∗,1} ⊗_{O_{F}} F, q_{∗,1}) .

Since the quadratic spaces are defined over O_{F} and become isomorphic over F and since the forms are unimodular, the spaces are isomorphic over O_{F} by lemma 4.4, i.e. there exists g′_{R} ∈ SO_{2n+2}(O_{F}) inducing isomorphisms

(M_{±,2}, q_{±,2}) ˜→ (M_{±,1}, q_{±,1}) and g′_{R,∗} : (M_{∗,2}, q_{∗,2}) ˜→ (M_{∗,1}, q_{∗,1}) .

If γ_{∗,i} denotes the restriction of γ_{i} to M_{∗,i} we get

(i)  γ_{∗,2} = g_{F,∗}^{−1} · γ_{∗,1} · g_{F,∗}  and
(ii)  γ_{∗,3} : = g′_{R,∗} · γ_{∗,2} · (g′_{R,∗})^{−1} ∈ SO(q_{∗,1}, O_{F}).

We have g_{F,∗} · (g′_{R,∗})^{−1} ∈ SO(q_{∗,1}, F). Now it follows from (i), (ii) and lemma 4.10 that there exists g_{∗} ∈ SO(M_{∗,1}, q_{∗,1}) satisfying g_{∗} · γ_{∗,3} · g_{∗}^{−1} = γ_{∗,1}. Then g_{R} := (id_{M_{+,1}} × id_{M_{−,1}} × g_{∗}) · g′_{R} ∈ SO_{2n+2}(O_{F}) satisfies g_{R} · γ_{2} · g_{R}^{−1} = γ_{1}.

(b) Let us assume that g_{F} ∈ SO_{2n+2}(F̄) satisfies γ_{2} := g_{F}^{−1} · γ_{1} · g_{F} ∈ O_{2n+2}(O_{F})^{−}. We only know that the quadratic spaces become isomorphic over F̄, but we have the additional discriminant conditions Δ(q_{+,1}) · Δ(q_{−,1}) · Δ(q_{∗,1}) = Δ(q_{+,2}) · Δ(q_{−,2}) · Δ(q_{∗,2}) and Δ(q_{∗,1}) = Δ(p_{∗,1}) · det(γ_{∗,1} − γ_{∗,1}^{−1}) = 1 · det(g_{F,∗}(γ_{∗,2} − γ_{∗,2}^{−1})g_{F,∗}^{−1}) = Δ(p_{∗,2})·det(γ_{∗,2}−γ_{∗,2}^{−1}) = Δ(q_{∗,2}) in O_{F}^{×}/(O_{F}^{×})^{2}, where we use the fact that the p_{∗,i} := q_{∗,i} · (γ_{∗,i} − γ_{∗,i}^{−1}) are unimodular skew symmetric by lemma 8.2 and thus have square determinants. The isomorphy-typ of the quadratic spaces (M_{±,1}, q_{±,1}) , (M_{∗,1}, q_{∗,1}) being fixed this means that there are two choices for the equivalence class of q_{+,2} but the isomorphy-typ of the other quadratic spaces (M_{−,2}, q_{−,2}) and (M_{∗,2}, q_{∗,2}) are then uniquely determined. To construct γ′_{1} we change the quadratic forms q_{+,1} on M_{+,1} and q_{−,1} on M_{−,1} to the other isomorphy-typ but make no change for M_{∗,1}, consider an isomorphism of quadratic spaces ι : R^{2n+2} ˜→ M_{+,1} ⊕ M_{−,1} ⊕ M_{∗,1} with respect to these modified forms on M_{±,1}, M_{∗,1} and the standard form J′_{2n+2} on R^{2n+2}, and put finally γ′_{1} = ι^{−1} ∘ γ_{1} ∘ ι. The statement of (b) now follows as in part (a).

Lemma 8.6. In the notations of 8.3 let b ∈ O_{2n+2}(O_{F})^{−} be residually semisimple and β = 1_{2r_{+}} × (−1_{2r_{−}}) × b_{∗} ∈ Sp_{2n}(O_{F}) a representing element of N (b) with b_{∗} − b_{∗}^{−1} ∈ GL_{2g}(O_{F}). Assume we have BC-matching topologically unipotent elements u_{+} ∈ SO_{2r_{+}+1}(F) and v_{+} ∈ Sp_{2r_{+}}(F) resp. u_{−} ∈ SO_{2r_{−}+1}(F) and v_{−} ∈ Sp_{2r_{−}}(F) and an additional topologically unipotent element u_{∗} ∈ Cent(b_{∗}, SO(q_{∗}, F)) ≃ Cent(b_{∗}, Sp(p_{∗}, F)). We form the topologically unipotent elements u = u_{+} × u_{−} × u_{∗} ∈ Cent(b, SO_{2n+2}(F)) and v = v_{+} × v_{−} × u_{∗} ∈ Cent(β, Sp_{2n}(F)). Then the elements g := bu = ub ∈ O_{2n+2}(F)^{−} and γ := βv = vβ ∈ Sp_{2n}(F) match.

Proof: As in the proof of lemma 6.6 we work in the case F = F̄ and assume that g resp. γ lie in the diagonal tori. The same holds for the residually semisimple parts b resp. β and the topologically unipotent parts u and v. As the matching of b and β is already proved in 8.4 we only have to examine the topologically unipotent elements. We can arrange the diagonal matrices u_{±} ∈ SO(q_{±}, F) such that their middle entries 1 correspond to the eigenvectors t_{n+1} · e_{n+1} ± e_{n+2} ∈ M_{±}, which get lost by the construction of N_{±}. Then the claim follows immediately from the definition of BC-matching 1.12, example 1.16 and the constructions in the proof of proposition 8.4.

Remark 8.7. The surjectivity statement of Proposition 8.4(b) is not true if R is a field, for example a p-adic field F: Let Δ ∈ F^{∗} denote a non square and

β = (\begin{matrix}a_{1} & & & b_{1} \\ & a_{2} & b_{2} \\ & b_{2}Δ & a_{2} \\ b_{1}Δ & & & a_{1} \end{matrix}) ∈ Sp_{4}(F)  where

a_{i} = \frac{λ_{i}^{2} + Δ}{λ_{i}^{2} − Δ},  b_{i} = \frac{2 · λ_{i}}{λ_{i}^{2} − Δ}  for λ_{i} ∈ F^{∗}, i = 1, 2.

Then we have N_{∗} = N and β_{∗} = β for N = F^{4} and can compute

q_{∗} : = p_{∗} · (β_{∗} − β_{∗}^{−1})^{−1} = J_{4} · antidiag(2b_{1}, 2b_{2}, 2Δb_{2}, 2Δb_{1})^{−1}

= diag (\frac{1}{2b_{1}} , \frac{−1}{2b_{2}}, \frac{1}{2Δb_{2}} , \frac{−1}{2Δb_{1}} ) = \frac{−1}{2Δb_{1}} · diag (−Δ, Δ · \frac{b_{1}}{b_{2}} , −\frac{b_{1}}{b_{2}} , 1) .

Thus the quadratic form q_{∗} on N is anisotropic if b_{1} · b_{2}^{−1} is not a norm of the extension F \sqrt{Δ}/F. In this case (N, q_{∗}) cannot be obtained as direct summand of the six dimensional quadratic split space (F^{6}, J′_{6}). The considerations of 8.3 and 8.4(b) then show that the conjugacy class of β is not in the image of N. The following theorem is again the fundamental lemma for a stable endoscopic lift modulo the BC-conjecture. But the non surjectivity of N in the case of local fields forces us to include the vanishing statement for orbital integrals of elements, that do not match.

Theorem 8.8. Assume that conjecture (BC_{m}) is true for all m ≤ n.

(a) If g ∈ S̃O_{2n+2}(F) = O_{2n+2}(F) with det(g) = −1 and γ ∈ Sp_{2n}(F) are matching semisimple elements then we have

O_{g}^{st}(1, S̃O_{2n+2}) = O_{γ}^{st}(1, Sp_{2n}). (iii)

(b) If the semisimple γ ∈ Sp_{2n}(F) matches with no element of S̃O_{2n+2}(F), then we have O_{γ}^{st}(1, Sp_{2n}) = 0.

Proof: Since the proof of (a) is similar to the proofs of theorems 6.7 and 7.11 we will be sketchy in some steps. We remark that (b) is an immediate corollary of the considerations in Step 1: If the right hand side of (iii) does not vanish one can construct an element g ∈ S̃O_{2n+2}(F) matching with γ using proposition 8.4 and lemma 8.6.

\underline{Step 1} As in the cited proofs we may assume without loss of generality that g ∈ O_{2n+2}(O_{F}) and γ ∈ Sp_{2n}(O_{F}). We may furthermore assume that g = (u_{+}, u_{−}, u_{∗}) · s and γ = (v_{+}, v_{−}, u_{∗})·σ are the topological Jordan decompositions with BC-matching topologically unipotent u_{+} ∈ SO_{2r_{+}+1}(O_{F}) and v_{+} ∈ Sp_{2r_{+}}(O_{F}) respectively u_{−} ∈ SO_{2r_{−}+1}(O_{F}) and v_{−} ∈ Sp_{2r_{−}}(O_{F}), matching residually semisimple s ∈ O_{2n+2}(O_{F}) and σ ∈ Sp_{2n}(O_{F}) and topologically unipotent u_{∗} ∈ Cent(σ_{∗}, Sp_{2g})(O_{F}).

\underline{Step 2} As in 6.7 we obtain all relevant conjugacy classes in the stable conjugacy class of γ if we let v′_{+} resp. v′_{−} vary through a set of representatives for the conjugacy classes in the stable conjugacy class of v_{+} resp. v_{−} in Sp_{2r_{+}}(F) resp. Sp_{2r_{−}}(F) and u′_{∗} through a set of representatives for the conjugacy classes in the stable conjugacy class of u_{∗} in Cent(σ_{∗}, Sp_{2g})(F) and then consider all γ′ = σ · (v′_{+}, v′_{−}, u′_{∗}) i.e.

O_{γ}^{st}(1, Sp_{2n}) (iv)

= ∑_{v′_{+}∼v_{+}} O_{v′_{+}}(1, Sp_{2r_{+}}) · ∑_{v′_{−}∼v_{−}} O_{v′_{−}}(1, Sp_{2r_{−}}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(σ_{∗}, Sp_{2g})).

\underline{Step 3} In the Θ-twisted situation O_{2n+2} all relevant Θ-conjugacy classes are of the form g′ = s′ · (u′_{+}, u′_{−}, u′_{∗}) where u′_{∗} is as above, u′_{±} vary through a set of representatives for the conjugacy classes in the stable class of u_{±} ∈ SO_{2r_{±}+1}(F) and s′ is either s or a representative for the corresponding other conjugacy class s′′ as in lemma 8.5(b). Observe the centralizers SO_{2n+2}^{s} and SO_{2n+2}^{s′′} can be identified, since the two equivalence classes of symmetric unimodular forms on a free O_{F} -module of odd rank have representatives which are scalar multiples of each other. Therefore we can use the same collections of u′_{±} and u′_{∗} for s as for s′′. The appearance of s′′ thus introduces just an additional factor 2 in the computation. But since the centralizers SO_{2n+2}^{s} and SO_{2n+2}^{s′′} have two connected components, there appears an additional factor \frac{1}{2} when we apply the Kazhdan-lemma 5.5. Thus we get:

O_{g}^{st}(1, O_{2n+2}) = ∑_{(u′_{+},u′_{−},u′_{∗})∼(u_{+},u_{−},u_{∗})} O_{(u′_{+},u′_{−},u′_{∗})}(1, SO_{2n+2}^{s}) (v)

= ∑_{u′_{+}∼u_{+}}O_{u′_{+}}(1, SO_{2r_{+}+1}) · ∑_{u′_{−}∼u_{−}} O_{u′_{−}}(1, SO_{2r_{−}+1}) · ∑_{u′_{∗}∼u_{∗}} O_{u′_{∗}}(1, Cent(σ_{∗}, Sp_{2g})).

In the last step we applied lemma 5.7 in the situation G = SO_{2n+2}^{s} = G_{1} × {±1_{2n+2}} where G_{1} = SO_{2r_{+}+1} × SO_{2r_{−}+1} × Cent(σ_{∗}, Sp_{2g}).

\underline{Step 4} Since v_{±} and u_{±} are BC-matching we can apply (BC_{r_{±}}) to get that the right hand sides of (iv) and (v) coincide, which proves the theorem.

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Address of all authors: Mathematisches Institut Im Neuenheimer Feld 288 D– 69120 Heidelberg ballmann@mathi.uni-heidelberg.de weissauer@mathi.uni-heidelberg.de weselman@mathi.uni-heidelberg.de