ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 11–20 (March 4, 2004) S 1079-6762(04)00125-8

ENDOSCOPIC DECOMPOSITION OF CHARACTERS OF CERTAIN CUSPIDAL REPRESENTATIONS

DAVID KAZHDAN AND YAKOV VARSHAVSKY

(Communicated by Svetlana Katok)

Abstract. We construct an endoscopic decomposition for local L-packets associated to irreducible cuspidal Deligne-Lusztig representations. Moreover, the obtained decomposition is compatible with inner twistings.

1. Introduction

Let E be a local non-Archimedean field, with ring of integers O and residue field 𝔽_{q} of characteristic p. We denote by Γ ⊃ W ⊃ I the absolute Galois, the Weil, and the inertia groups of E. Let G be a reductive group over E, ^{L}G = Ĝ ⋊ W its complex Langlands dual group, and D(G(E)) the space of invariant distributions on G(E).

Every admissible homomorphism λ : W → ^{L}G (see [Ko1, § 10]) gives rise to a finite group S_{λ} := π_{0}(ZĜ(λ)/Z(Ĝ)^{Γ}), where ZĜ(λ) is the centralizer of λ(W) in Ĝ. Every conjugacy class κ of S_{λ} defines an endoscopic subspace D_{κ,λ}(G(E)) ⊂ D(G(E)). For simplicity, we will restrict ourselves to the elliptic case, where λ(W) does not lie in any proper Levi subgroup of ^{L}G.

Langlands conjectured that every elliptic λ corresponds to a finite set Π_{λ}, called an L-packet, of cuspidal irreducible representations of G(E). Moreover, the subspace D_{λ}(G(E)) ⊂ D(G(E)), generated by the characters {χ(π)}_{π∈Π_{λ}}, must have an endoscopic decomposition. More precisely, it is expected ([La1, IV, 2]) that there exists a basis {a_{π}}_{π∈Π_{λ}} of the space of central functions on S_{λ} such that χ_{κ,λ} := ∑_{π∈Π_{λ}} a_{π}(κ)χ(π) belongs to D_{κ,λ}(G) for every conjugacy class κ of S_{λ}.

The goal of this paper is to construct the endoscopic decomposition of D_{λ}(G(E)) for tamely ramified λ’s such that ZĜ(λ(I)) is a maximal torus. In this case, G splits over an unramified extension of E, and λ factors through ^{L}T ↪ ^{L}G for an elliptic unramified maximal torus T of G. By the local Langlands correspondence for tori ([La2]), a homomorphism λ : W → ^{L}T defines a tamely ramified homomorphism θ : T (E) → ℂ^{×}. Each κ ∈ S_{λ} = T̂^{Γ}/Z(Ĝ)^{Γ} gives rise to an elliptic endoscopic datum E_{κ,λ} of G, while the characters of S_{λ} are in bijection with the conjugacy classes of embeddings T ↪ G, stably conjugate to the inclusion. Therefore each

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Received by the editors September 18, 2003 and, in revised form, January 19, 2004.

2000 Mathematics Subject Classification. Primary 22E50; Secondary 22E35.

Key words and phrases. Endoscopy, Deligne-Lusztig representations.

The work of the second author was supported by the Israel Science Foundation (Grant No.

38/01-1).

character a of S_{λ} gives rise to an irreducible cuspidal representation π_{a,λ} of G(E) (denoted by π_{a,θ} in Notation 2.3).

Our main result asserts, for fields E of sufficiently large residual characteristic, that each χ_{κ,λ} := ∑_{a} a(κ)χ(π_{a,λ}) is E_{κ,λ}-stable. Moreover, the resulting endoscopic decomposition of D_{λ}(G(E)) is compatible with inner twistings. For simplicity, we restrict ourselves to local fields of characteristic zero, while the case of positive characteristic follows by approximation (see [Ka3], [De]).

Our argument goes as follows. First we prove the stability of the restriction of χ_{κ,λ} to the subset of topologically unipotent elements of G(E). If p is sufficiently large, this assertion reduces to the analogous assertion about distributions on the Lie algebra. Now the stability follows from a combination of a Springer hypothesis [Ka1] and a generalization of a theorem of Waldspurger [Wa]. To prove the result in general, we use the topological Jordan decomposition ([Ka2]).

When this work was in the process of writing, we have heard that S. DeBacker and M. Reeder obtained similar results.

Notation and conventions. In addition to the notation introduced above, we use the following conventions:

For a reductive group G, always assumed to be connected, we denote by Z(G), G^{ad}, G^{der}, G^{sc}, G_{δ}, and G^{sr} the center of G, the adjoint group of G, the derived group of G, the simply connected covering of G^{der}, the centralizer of δ ∈ G, and the set of strongly regular semisimple elements of G (that is, the set of δ ∈ G such that G_{δ} ⊂ G is a maximal torus), respectively.

Denote by G, T , and L the Lie algebras of the algebraic groups G, T, and L. Let E be a local non-Archimedean field of characteristic zero, \overline{E} a fixed algebraic

closure of E, and E^{nr} a maximal unramified extension of E in \overline{E}.

For a reductive group G (resp. its Lie algebra G) over E, we denote by S(G(E)) (resp. S(G(E))) the space of locally constant measures with compact support. We denote by D(G(E)) (resp. D(G(E))) the space of invariant distributions on G(E) (resp. G(E)), namely G(E)-invariant linear functionals on S(G(E)) (resp. S(G(E))), where G(E) acts by conjugation. Whenever necessary we equip G(E) and G(E) with invariant measures, denoted by μ, defined by a translation-invariant top degree differential form on G. We denote by G(E)_{tu} (resp. G(E)_{tn}) the set of topologically unipotent (resp. topologically nilpotent) elements of G(E) (resp. G(E)). Finally, we denote by rk(G) the rank of G over E, and put e(G) := (−1)^{rk(G^{ad})}. Note that our sign e(G) differs from that defined by Kottwitz.

2. Formulation of the main result

2.1. Let L be a connected reductive group over 𝔽_{q}, and \overline{a} : \overline{T} ↪ L an embedding of a maximal elliptic torus of L. Following Deligne and Lusztig, we associate to every character \overline{θ} : \overline{T} (𝔽_{q}) → ℂ^{×} in general position an irreducible cuspidal representation ρ_{\overline{a},\overline{θ}} of L(𝔽_{q}) (see [DL, Prop. 7.4 and Thm. 8.3]).

2.2. There is an equivalence of categories T ↦ \overline{T} between tori over E splitting over E^{nr} and tori over 𝔽_{q}. Every such T has a canonical O-structure.

Notation 2.3. a) Let G be a reductive group over E, T a torus over E splitting over E^{nr}, and a : T ↪ G an embedding of a maximal elliptic torus of G. Then G splits over E^{nr}, and a(T (O)) lies in a unique parahoric subgroup G_{a} of G(E). Let G_{a^{+}} be the pro-unipotent radical of G_{a}. Then there exists a canonical reductive group L_{a} over 𝔽_{q} with an identification L_{a}(𝔽_{q}) = G_{a}/G_{a^{+}}. Moreover, a : T ↪ G induces an embedding \overline{a} : \overline{T} ↪ L_{a} of a maximal elliptic torus of L_{a}.

b) Let θ : T (E) → ℂ^{×} be a character in general position, trivial on Ker[T (O) → \overline{T} (𝔽_{q})]. Denote by \overline{θ} : \overline{T} (𝔽_{q}) → ℂ^{×} the character of \overline{T} (𝔽_{q}) defined by θ. Then there exists a unique irreducible representation ρ_{a,θ} of Z(G)(E)G_{a}, whose central character is the restriction of θ, extending the inflation to G_{a} of the cuspidal Deligne-Lusztig representation ρ_{\overline{a},\overline{θ}} of L_{a}(𝔽_{q}). We denote by π_{a,θ} the induced cus-

pidal representation Ind_{Z(G)(E)G_{a}}^{G(E)} ρ_{a,θ} of G(E).

2.4. Recall (see [Ko2, Thm 1.2]) that for every reductive group G over E, H^{1}(E, G) is canonically isomorphic to the group π_{0}(Z(Ĝ)^{Γ})^{D} of characters of π_{0}(Z(Ĝ)^{Γ}). If T is a maximal torus of G, we get a commutative diagram:

H^{1}(E, T) ⟶^{∼} π_{0}(T̂^{Γ})^{D}
↓  ↓
H^{1}(E, G) ⟶^{∼} π_{0}(Z(Ĝ)^{Γ})^{D}.

In particular, we have a canonical surjection

T̂^{Γ}/Z(Ĝ)^{Γ} → Coker[π_{0}(Z(Ĝ)^{Γ}) → π_{0}(T̂^{Γ})] →^{∼} (Ker [H^{1}(E, T) → H^{1}(E, G)])^{D}.

Notation 2.5. a) To every pair a, a′ of stably conjugate embeddings T ↪ G, one associates the class inv(a′, a) ∈ Ker [H^{1}(E, T) → H^{1}(E, G)]. This is the class of a cocycle c_{σ} = g^{−1}σ(g), where g ∈ G(\overline{E}) is such that a′ = gag^{−1} (compare [Ko2, 4.1]).

b) To each κ ∈ T̂^{Γ}/Z(Ĝ)^{Γ}, an embedding a_{0} : T ↪ G, and a character θ of T (E) as in Notation 2.3, we associate the invariant distribution

χ_{a_{0},κ,θ} := e(G) ∑_{a}⟨inv(a, a_{0}), κ⟩χ(π_{a,θ}).

Here a runs over a set of representatives of conjugacy classes of embeddings which are stably conjugate to a_{0}, and χ(π_{a,θ}) denotes the character of π_{a,θ}.

Notation 2.6. Each pair (a, κ), where a : T ↪ G is an embedding of a maximal torus of G and κ is an element of T̂^{Γ}, gives rise to an isomorphism class E_{(a,κ)} of an endoscopic datum of G. Furthermore, E_{(a,κ)} is elliptic if a(T) is an elliptic torus of G (see [Ko1, §7] for the definitions of endoscopic data, and compare [La1, II, 4]).

More precisely, each embedding η : T̂ ↪ Ĝ, whose conjugacy class corresponds to the stable conjugacy class of a, defines an endoscopic datum E_{(a,κ,η)} = (s, ρ), consisting of a semisimple element s = η(κ) of Ĝ and a homomorphism ρ : Γ ⟶^{ρ_{T}}

Norm_{Aut Ĝ}(η(T̂))_{s}/η(T̂) ⟶^{ρ′} Out(Ĝ_{s}^{0}). Here ρ_{T} is induced by the E-structure of T, and ρ′ is induced by the inclusion Norm_{Aut Ĝ}(η(T̂))_{s} ⊂ Norm_{Aut Ĝ}(Ĝ_{s}^{0}). Moreover, the isomorphism class of E_{(a,κ,η)}, denoted by E_{(a,κ)}, does not depend on η.

Notation 2.7. For each γ ∈ G^{sr}(E) and ξ ∈ Ĝ_{γ}^{Γ},

(i) put E_{(γ,ξ)} := E_{(a_{γ},ξ)}, where a_{γ} : G_{γ} ↪ G is the inclusion map;

(ii) fix an invariant measure dg_{γ} on G_{γ}(E), and put

O_{γ}(ϕ) := ∫_{G(E)/G_{γ}(E)} f(gγg^{−1})\frac{dg}{dg_{γ}}

for each ϕ = fdg ∈ S(G(E));

(iii) denote by \overline{ξ} ∈ π_{0}(Ĝ_{γ}^{Γ}/Z(Ĝ)^{Γ}) the class of ξ; (iv) denote by O\overline{ξ} ; γ ∈ D(G(E)) the sum ∑_{γ′}⟨inv(γ′, γ), \overline{ξ}⟩O_{γ′}, taken over a set of

representatives of the conjugacy classes stably conjugate to γ, where each dg_{γ′} is compatible with dg_{γ}.

Definition 2.8. Let E be an endoscopic datum of G.

(i) A measure ϕ ∈ S(G(E)) is called E-unstable if O\overline{ξ} ; γ(ϕ) = 0 for all pairs (γ, ξ) as in Notation 2.7 for which E_{(γ,ξ)} is isomorphic to E.

(ii) A distribution F ∈ D(G(E)) is called E-stable if F (ϕ) = 0 for all E-unstable ϕ ∈ S(G(E)).

Theorem 2.9. Assume that p > dim G^{der}. Then for each triple (a_{0}, κ, θ), the distribution χ_{a_{0},κ,θ} is E_{(a_{0},κ)}-stable.

Notation 2.10. For an endoscopic datum E = (s, ρ), choose a representative s̃ ∈ Ĝ^{sc} of s, and let Z(E) be the set of z ∈ Z(Ĝ^{sc})^{Γ} for which there exists g ∈ Ĝ_{s} commuting with ρ : Γ → Out(Ĝ_{s}^{0}) such that gs̃g^{−1} = zs̃. Then Z(E) is a subgroup of Z(Ĝ^{sc})^{Γ}, depending only on the isomorphism class of E.

Definition 2.11. Let E be an endoscopic datum of G. An inner twisting φ : G → G′ is called E-admissible if the corresponding class inv(G′, G) ∈ H^{1}(E, G^{ad}) ≅ (Z(Ĝ^{sc})^{Γ})^{D} is orthogonal to Z(E) ⊂ Z(Ĝ^{sc})^{Γ}.

Definition 2.12. Let G be a reductive group over E, E = (s, ρ) an elliptic endoscopic datum of G, and φ : G → G′ an E-admissible inner twisting. Fix a triple (a, a′; κ), consisting of a pair a : T ↪ G and a′ : T ↪ G′ of stably conjugate embeddings of maximal tori, and an element κ ∈ T̂^{Γ} such that E_{(a,κ)} ≅ E.

a) Consider ϕ ∈ S(G(E)) and ϕ′ ∈ S(G′(E)). They are called (a, a′; κ)-indistin-guishable if they satisfy the following conditions.

(A) For every γ ∈ G^{sr}(E) and ξ ∈ Ĝ_{γ}^{Γ} such that E_{(γ,ξ)} ≅ E and O\overline{ξ} ; γ(ϕ) ≠ 0, (i) there exists γ′ ∈ G′(E) stably conjugate to γ;

(ii) we have O\overline{ξ} ; γ′(ϕ′) = ⟨\frac{γ′,γ;ξ}{a′,a;κ}⟩O\overline{ξ} ; γ(ϕ).

Here ⟨\frac{γ′,γ;ξ}{a′,a;κ}⟩ ∈ ℂ^{×} is the invariant ⟨\frac{a_{γ′} ,a_{γ};ξ}{a′,a;κ} ⟩ defined in the Appendix for embeddings a_{γ} : G_{γ} ↪ G and a_{γ′} : G_{γ} ↪ G′ such that a_{γ}(γ) = γ and a_{γ′}(γ) = γ′.

(B) Condition (A) holds if G, a_{0}, γ, ϕ are interchanged with G′, a′_{0}, γ′, ϕ′.

b) The distributions F ∈ D(G(E)) and F′ ∈ D(G′(E)) are called (a, a′; κ)- equivalent if F (ϕ) = F′(ϕ′) for every two (a, a′; κ)-indistinguishable measures ϕ and ϕ′.

Remark 2.13. If ϕ is E_{(a_{0},κ)}-unstable, then ϕ and ϕ′ = 0 are (a_{0}, a′_{0}; κ)-indistin-guishable. Therefore every two (a_{0}, a′_{0}; κ)-equivalent distributions F and F′ are E_{(a_{0},κ)}-stable.

Main Theorem 2.14. Assume that p > dim G^{der}. Let φ : G → G′ be an E_{(a_{0},κ)}- admissible inner twisting. Let a′_{0} : T ↪ G′ be an embedding which is stably conjugate to a_{0}. Then the distributions χ_{a_{0},κ,θ} on G(E) and χ_{a′_{0},κ,θ} on G′(E) are (a_{0}, a′_{0}; κ)-equivalent.

Remark 2.15. a) By Remark 2.13, Theorem 2.9 follows from the Main Theorem. b) We believe that a much smaller bound on p would suffice.

3. Basic ingredients of the argument

3.1. A generalization of a theorem of Waldspurger. Suppose that we are in the situation of Definition 2.12. Then φ induces an inner twisting G → G′. As in Definition 2.12, one can define (a, a′; κ)-equivalence of F ∈ D(G(E)) and F′ ∈ D(G′(E)).

Fix a nontrivial character ψ : E → ℂ^{×}, a nondegenerate G-invariant pairing ⟨·, ·⟩ on G, and φ-compatible invariant measures on G(E) and G′(E). Then φ defines a nondegenerate G′-invariant pairing ⟨·, ·⟩′ on G′. These data determine the Fourier transforms F ↦ F(F) on G(E) and G′(E).

Theorem 3.1. The distributions F ∈ D(G(E)) and F′ ∈ D(G′(E)) are (a, a′; κ)- equivalent if and only if e(G)F(F) and e(G′)F(F′) are (a, a′; κ)-equivalent.

The proof is a generalization of that of Waldspurger [Wa], who treated the case ϕ′ = 0 (compare also [KP, Thm. 2.7.1], where the stable case is considered).

3.2. Springer hypothesis. In the notation of 2.1, assume that \overline{a}(\overline{T} )(𝔽_{q}) ⊂ L(𝔽_{q}) contains an L-regular element \overline{t} [and that p is so large that the logarithm defines an isomorphism log : L_{un} →^{∼} L_{nil} between unipotent elements of L and nilpotent elements of L]. Let δ\overline{t} be the characteristic function of the Ad(L(𝔽_{q}))-orbit of \overline{t}, and let F(δ\overline{t}) be its Fourier transform. We need the following result of [Ka1].

Theorem 3.2. For every u ∈ L_{un}(𝔽_{q}), we have

Tr ρ_{\overline{a},\overline{θ}}(u) = q^{−(dim L−dim \overline{T})/2}F(δ\overline{t})(log(u)).

3.3. Topological Jordan decomposition. We will call an element γ ∈ G(E) compact if it generates a relatively compact subgroup of G(E). We will call an element γ ∈ G(E) topologically unipotent if the sequence {γ^{p^{n}}}_{n} converges to 1. Every topologically unipotent element is compact. The following result is a rather straightforward generalization of [Ka2, Lem. 2, p. 226].

Lemma 3.3. For every compact element γ ∈ G(E) there exists a unique decomposition γ = δu such that δ and u commute, δ is of finite order prime to p, and u is topologically unipotent. In particular, this decomposition is compatible with conjugation and field extensions.

4. A sketch of the proof of the main theorem

4.1. Reformulation of the problem.

Notation 4.1. To each a : T ↪ G and θ : T (E) → ℂ^{×} as in Notation 2.3 we associate a function t_{a,θ} on G(E) supported on Z(G)(E)G_{a} and equal to Tr ρ_{a,θ} there. Since t_{a,θ} is cuspidal, the integral

F_{a,θ}(γ) := \frac{1}{μ((G^{ad})_{a})} ∫_{G(E)/Z(G)(E)} t_{a,θ}(gγg^{−1})dg

stabilizes for every γ ∈ G^{sr}(E) (see [HC, Lem. 23]), thus providing us with a locally constant invariant function F_{a,θ} on G^{sr}(E).

Lemma 4.2. For each a and θ, F_{a,θ} is a locally L^{1}-function on G(E). Moreover, the corresponding distribution equals χ(π_{a,θ}).

Proof. The assertion follows from Harish-Chandra’s theorem [HC, Thm. 16]. □

Notation 4.3. For every γ_{0} ∈ G^{sr} and \overline{ξ} ∈ π_{0}(Ĝ_{γ_{0}}^{Γ}/Z(Ĝ)^{Γ}) we define

(4.1)  Σ_{G;γ_{0},\overline{ξ};a_{0},κ} := e(G) ∑_{a} ∑_{γ}⟨inv(a, a_{0}), κ⟩⟨inv(γ, γ_{0}), \overline{ξ}⟩^{−1}F_{a,θ}(γ),

where a and γ run over sets of representatives of the conjugacy classes within the stable conjugacy classes of a_{0} and γ_{0}, respectively.

Theorem 4.4. For all γ_{0} ∈ G^{sr} and \overline{ξ} ∈ π_{0}(Ĝ_{γ_{0}}^{Γ}/Z(Ĝ)^{Γ}) such that Σ_{G;γ_{0},\overline{ξ};a_{0},κ} ≠ 0,

(i) there exists a representative ξ ∈ Ĝ_{γ_{0}}^{Γ} of \overline{ξ} such that E_{γ_{0},ξ} ≅ E_{(a_{0},κ)}; (ii) if φ : G → G′ is (E, a_{γ_{0}}, \overline{ξ})-admissible (see Definition A.1), then for every

ξ as in (i) and every stably conjugate γ′_{0} ∈ G′(E) of γ_{0} we have

Σ_{G′;γ′_{0},\overline{ξ};a′_{0},κ} = ⟨\frac{γ′_{0}, γ_{0}; ξ}{a′, a; κ} ⟩Σ_{G;γ_{0},\overline{ξ};a_{0},κ}.

4.5. It follows from Lemma 4.2 that Theorem 4.4 is equivalent to the Main Theorem. Moreover, by standard arguments, Theorem 4.4 reduces to the case when the derived group of G is simply connected.

4.6. From now on we will assume that G^{der} = G^{sc}. In particular, the centralizer of each semisimple element of G is connected, and each G_{a} is a maximal compact subgroup of G(E). We fix (γ_{0}, \overline{ξ}) such that Σ_{G;γ_{0},\overline{ξ};a_{0},κ} ≠ 0. Since Σ_{G;zγ_{0},\overline{ξ};a_{0},κ} = θ(z)Σ_{G;γ_{0},\overline{ξ};a_{0},κ} for each z ∈ Z(G)(E) and since the support of each t_{a,θ} consists of elements compact modulo center, we can assume that γ_{0} is compact with topological Jordan decomposition γ_{0} = δ_{0}u_{0}. Moreover, we can assume that either γ_{0} is topologically unipotent, or δ_{0} ∉ Z(G)(E).

4.2. The topologically unipotent case.

4.7. Since p does not divide the order of Z(G^{der}), the canonical map G^{der}(E)_{tu} × Z(G)(E)_{tu} → G(E)_{tu} is an isomorphism. Therefore to prove Theorem 4.4 for topologically unipotent γ_{0}, we can assume that G is semisimple and simply connected.

Notation 4.8. Denote by Φ_{G} : G → G the composition map

G ⟶^{Ad} GL(G) ⟶^{log_{(p)}} End(G) ⟶^{pr} G,

where log_{(p)}(1 − A) = − ∑_{i=1}^{p−1} \frac{A^{i}}{i}, and pr is the canonical projection, defined by the standard pairing (A, B) ↦ Tr AB on End(G).

Lemma 4.9. The map Φ_{G} defines a G(E)-equivariant homeomorphism

G(E)_{tu} →^{∼} G(E)_{tn} ,

where G(E) acts by conjugation. Moreover, for every parahoric subgroup G_{x} of G(E), Φ_{G} induces a bijection (Φ_{G})_{x} : (G_{x})_{tu} →^{∼} (G_{x})_{tn}, which in turn induces the logarithm map log : (L_{x})_{un}(𝔽_{q}) →^{∼} (L_{x})_{nil}(𝔽_{q}).

Notation 4.10. a) By our assumption on p, there exists t ∈ T (O) whose reduction \overline{t} ∈ \overline{T} (𝔽_{q}) is not fixed by any nontrivial element of the Weyl group of G.

b) For every a : T ↪ G as in Notation 2.3, we denote by Ω_{a,t} ⊂ L_{a}(𝔽_{q}) the Ad(L_{a}(𝔽_{q}))-orbit of \overline{a}(\overline{t}), by Ω̃_{a,t} ⊂ G_{a} ⊂ G the preimage of Ω_{a,t}, and let δ_{a,t} be the characteristic function of Ω̃_{a,t}.

c) As the centralizer G_{y} of each y ∈ Ω̃_{a,t} is G_{a}-conjugate to a(T), the integral

Δ_{a,t}(x) := \frac{1}{μ(G_{a})} ∫_{G(E)} δ_{a,t}(Ad(g)x)dg

converges absolutely for each x ∈ G(E). Thus it defines an element of D(G(E)). Similarly to Notation 2.5, we consider Δ_{a_{0},κ,t} := e(G) ∑_{a}⟨inv(a, a_{0}), κ⟩Δ_{a,t} ∈ D(G(E)).

Lemma 4.11. Let I^{+} ⊂ G(E) be a maximal topologically nilpotent subalgebra. Assume that ψ : E → ℂ^{×} is trivial on the maximal ideal M ⊂ O and induces a nontrivial character of 𝔽_{q}. Then for each u ∈ G(E)_{tu} we have

t_{a,θ}(u) = μ(I^{+})^{−1}F(δ_{a,t})(Φ_{G}(u)).

Proof. The assumption on ψ implies that G_{a^{+}} is the orthogonal complement of G_{a} with respect to the pairing (x, y) ↦ ψ(⟨x, y⟩). Therefore our lemma is an immediate consequence of the definition of the Fourier transform (over E and 𝔽_{q}), Theorem 3.2, Lemma 4.9, and the equality q^{(dim L_{a}−dim \overline{T})/2}μ(G_{a^{+}}) = μ(I^{+}). □

4.12. Now we are ready to show that (χ_{a_{0},κ,θ})_{|G(E)_{tu}} and (χ_{a′_{0},κ,θ})_{|G′(E)_{tu}} are (a_{0}, a′_{0}; κ)-equivalent. First of all, by direct calculations, e(G)Δ_{a_{0},κ,t} is (a_{0}, a′_{0}; κ)- equivalent to e(G′)Δ_{a′_{0},κ,t}. Hence, by Theorem 3.1, F(Δ_{a_{0},κ,t}) is (a_{0}, a′_{0}; κ)-equiv-alent to F(Δ_{a′_{0},κ,t}). Using Lemmas 4.2, 4.9, and 4.11 we see that χ_{a_{0},κ,θ} has the same restriction to G(E)_{tu} as μ(I^{+})^{−1}Φ_{G}^{∗}(F(Δ_{a_{0},κ,t})), and similarly for χ_{a′_{0},κ,θ} and μ(I^{′+})^{−1}Φ_{G′}^{∗}(F(Δ_{a′_{0},κ,t})). Since Φ_{G} is a G^{ad}-invariant algebraic morphism defined over E, the assertion follows from the equality μ(I^{+}) = μ(I^{′+}).

4.3. The general case. It remains to prove Theorem 4.4 for δ_{0} ∉ Z(G)(E) (see 4.6). We are going to deduce the assertion from that for G_{δ_{0}}.

Proposition 4.13. For every embedding a : T ↪ G and a compact element γ in G(E) with topological Jordan decomposition γ = δu, we have

e(G)F_{a,θ}(γ) = e(G_{δ}) ∑_{b} θ(b^{−1}(δ))F_{b,θ}(u).

Here b runs over the set of conjugacy classes of embeddings b : T ↪ G_{δ} whose composition with the inclusion G_{δ} ⊂ G is conjugate to a.

Proof. The proposition follows by direct calculation from the recursive formula ([DL, Thm. 4.2]) for characters of Deligne-Lusztig representations. □

Notation 4.14. a) We say that t ∈ T (E) is (G, a_{0}, γ_{0})-relevant if there exists an embedding b_{0} : T ↪ G_{δ_{0}} ⊂ G stably conjugate to a_{0} such that b(t) = δ_{0}.

b) Assume that t ∈ T (E) is (G, a_{0}, γ_{0})-relevant. Since a_{0}(T) ⊂ G is elliptic, for each δ ∈ G(E) stably conjugate to δ_{0} there exists an embedding b_{t,δ} : T ↪ G_{δ} ⊂ G stably conjugate to a_{0} such that b_{t,δ}(t) = δ. Further, b_{t,δ} is unique up to stable conjugacy, and the endoscopic datum E_{t,κ} := E_{(b_{δ,t},κ)} of G_{δ_{0}} is independent of δ.

c) We will write δ_{1} ∼_{E_{t,κ}} δ (resp. δ′ ∼_{E_{t,κ}} δ) if δ, δ_{1} ∈ G(E) (resp. δ ∈ G(E) and δ′ ∈ G′(E)) are stably conjugate to δ_{0}, and G_{δ_{1}} (resp. G′_{δ′}) is an E_{t,κ}-admissible inner form of G_{δ} (see Definition 2.11).

4.15. Using Proposition 4.13, we see that

(4.2)  Σ_{G;γ_{0},\overline{ξ};a_{0},κ} = ∑_{t} θ(t) ∑_{δ} I_{t,δ},

where (i) t runs over the set of (G, a_{0}, γ_{0})-relevant elements of T (E); (ii) δ runs over a set of representatives of the conjugacy classes within the stable

conjugacy class of δ_{0}; (iii) I_{t,δ} vanishes unless there exists an element γ ∈ G(E) stably conjugate to

γ_{0} with topological Jordan decomposition γ = δu, in which case we get

(4.3)  I_{t,δ} = ⟨inv(b_{t,δ}, a_{0}), κ⟩⟨inv(γ, γ_{0}), \overline{ξ}⟩^{−1}Σ_{G_{δ};u,\overline{ξ};b_{t,δ},κ}.

4.16. For simplicity of the exposition, we will restrict ourselves to the case when γ_{0} ∈ G(E) is elliptic. Choose t which has a nonzero contribution to (4.2). Replacing δ_{0} by a stably conjugate element we can assume that ∑_{δ∼_{E_{t,κ}} δ_{0}} I_{t,δ} ≠ 0 and I_{t,δ_{0}} ≠ 0. So Σ_{G_{δ_{0}} ;u_{0},\overline{ξ};b_{t,δ},κ} ≠ 0. Hence by Theorem 4.4 for G_{δ_{0}} there exists a representative

ξ ∈ Ĝ_{γ_{0}}^{Γ} of \overline{ξ} such that the endoscopic datum E_{(u_{0},ξ)} of G_{δ_{0}} is isomorphic to E_{t,κ}. Therefore there exist embeddings η_{1} : Ĝ_{γ_{0}} ↪ Ĝ_{δ_{0}} and η_{2} : T̂ ↪ Ĝ_{δ_{0}} such that E_{(γ_{0},ξ,η_{1})} = E_{(b_{t,δ_{0}} ,κ,η_{2})} (compare Notation 2.6) and η_{2}(κ) = zη_{1}(ξ) for a certain z ∈ Z(Ĝ_{δ_{0}})^{Γ}. Moreover, z is defined up to multiplication by an element of Z(E_{t,κ}). Therefore for all δ ∼_{E_{t,κ}} δ_{0}, the expression ⟨inv(δ, δ_{0}), z⟩ is independent of the choice of the η_{i}’s.

Claim 4.17. For each δ ∼_{E_{t,κ}} δ_{0} we have I_{t,δ} = ⟨inv(δ, δ_{0}), z⟩I_{t,δ_{0}}.

Proof. Since Σ_{G_{δ_{0}} ;u_{0},\overline{ξ};b_{t,δ},κ} ≠ 0, Theorem 4.4 for inner forms G_{δ} and G_{δ_{0}} implies that for every stably conjugate u ∈ G_{δ}(E) of u_{0} ∈ G_{δ_{0}}(E), we have

Σ_{G_{δ};u,\overline{ξ};b_{t,δ},κ} = ⟨\frac{u, u_{0}; ξ}{b_{t,δ}, b_{t,δ_{0}}; κ}⟩Σ_{G_{δ_{0}} ;u_{0},\overline{ξ};b_{t,δ_{0}} ,κ} .

Then γ := δu ∈ G(E) is stably conjugate to γ_{0}, and the assertion follows by direct calculation from (4.3). □

4.18. Now we are ready to show the validity of (i), (ii) of Theorem 4.4. (i) As ∑_{δ∼_{E_{t,κ}} δ_{0}} I_{t,δ} ≠ 0, we get from Claim 4.17 that ∑_{δ∼_{E_{t,κ}} δ_{0}}⟨inv(δ, δ_{0}), z⟩ ≠

0. By the definition of E_{t,κ}-equivalence, this implies that z belongs to Z(E_{t,κ})Z(Ĝ)^{Γ}. Thus changing η_{1} (or η_{2}), we can assume that z ∈ Z(Ĝ)^{Γ}. Since (γ_{0}, ξ) and (b_{t,δ_{0}}, κ) define isomorphic endoscopic data of G_{δ_{0}}, we therefore conclude that E_{(γ_{0},ξ)} ≅ E_{(a_{0},κ)}.

(ii) Since T is elliptic, an element t ∈ T (E) is (G, a_{0}, γ_{0})-relevant if and only if it is (G′, a′_{0}, γ′_{0})-relevant. Thus it will suffice to show that for every such t we have ∑_{δ′} I_{t,δ′} = ⟨\frac{γ′_{0},γ_{0};ξ}{a′_{0},a_{0};κ}⟩ ∑_{δ} I_{t,δ}. For every stably conjugate δ ∈ G(E) of δ_{0}, there exists a stably conjugate δ′ ∈ G′(E) of δ′_{0} such that δ′ ∼_{E_{t,κ}} δ. Therefore it will suffice to show that for every such pair δ′ ∼_{E_{t,κ}} δ, we have I_{t,δ′} = ⟨\frac{γ′_{0},γ_{0};ξ}{a′_{0},a_{0};κ}⟩I_{t,δ}. The latter equality can be proved by the same arguments as Claim 4.17.

Appendix A

Let G be a reductive group over E, E = (s, ρ) an elliptic endoscopic datum of G, and φ : G → G′ an E-admissible inner twisting. For every two triples (a′_{i}, a_{i}; κ_{i}), i = 1, 2, where a_{i} : T_{i} ↪ G and a′_{i} : T_{i} ↪ G′ are stably conjugate embeddings of

maximal tori, and κ_{i} is an element of T̂_{i}^{Γ} such that E_{(a_{i},κ_{i})} is isomorphic to E, we are going to define an invariant ⟨\frac{a′_{1},a_{1};κ_{1}}{a′_{2},a_{2};κ_{2}} ⟩ ∈ ℂ^{×}.

Step 1. Replacing G, G′, T_{i}, κ_{i}, and E by G^{sc}, G^{′ sc}, T_{i}^{sc} := a_{i}^{−1}(G^{sc}) = a_{i}^{′−1}(G^{′ sc}),

the image of κ_{i} in T̂_{i}^{sc}^{Γ}, and the corresponding endoscopic datum of G^{sc}, respectively, we can assume that G is semisimple and simply connected. Let T_{1,2} be the quotient of the product T_{1} × T_{2} by the subgroup {(z, z^{−1})|z ∈ Z(G) = Z(G′)}.

Step 2. Choose elements g_{1}, g_{2}, and \{c̃_{σ}\}_{σ∈Γ} of G(\overline{E}) such that a′_{i} = φ(g_{i}a_{i}g_{i}^{−1}) and each c̃_{σ} is a representative of φ^{−1}^{σ}φ ∈ G^{ad}(\overline{E}). Then each g_{i}^{−1}c̃_{σ}^{σ}g_{i} ∈ G(\overline{E}) belongs to a_{i}(T_{i}(\overline{E})), and the images of (a_{1}^{−1}(g_{1}^{−1}c̃_{σ}^{σ}g_{1}), a_{2}^{−1}((g_{2}^{−1}c̃_{σ}^{σ}g_{2})^{−1})) in T_{1,2}(\overline{E}) form a cocycle, whose cohomology class inv(\frac{a′_{1},a_{1}}{a′_{2},a_{2}} ) ∈ H^{1}(E, T_{1,2}) is independent of the choices.

Step 3. Choose embeddings η_{i} : T̂_{i} ↪ Ĝ such that E_{(a_{i},κ_{i},η_{i})} = (s, ρ) and a representative s̃ ∈ Ĝ^{sc} = Ĝ^{ad} of s. Put T_{i}^{ad} := T_{i}/a_{i}^{−1}(Z(G)). Each η_{i} defines an embedding η̃_{i} : T̂_{i}^{ad} ↪ Ĝ^{ad}, hence an element κ̃_{i} = κ(s̃, η_{i}) := η_{i}^{−1}(s̃) ∈ T̂_{i}^{ad}. Then the image of (κ̃_{1}, κ̃_{2}) in T̂_{1}^{ad} × T̂_{2}^{ad}/Z(Ĝ^{ad}) ≅ T̂_{1,2}, denoted by κ_{1,2}, is Γinvariant. Moreover, as φ : G → G′ is E-admissible, the expression ⟨\frac{a′_{1},a_{1};κ_{1}}{a′_{2},a_{2};κ_{2}} ⟩ :=

⟨inv(\frac{a′_{1},a_{1}}{a′_{2},a_{2}} ), κ_{1,2}⟩ ∈ ℂ^{×} is independent of the choices.

Definition A.1. Let E = (s, ρ) be an endoscopic datum of G, φ : G → G′ an inner twisting, a : T ↪ G an embedding of a maximal torus, and κ an element of T̂^{Γ} such that E_{(a,κ)} ≅ E. We say that φ : G → G′ is (E, a, \overline{κ})-admissible, if for all representatives κ′ ∈ T̂^{Γ} of \overline{κ} ∈ π_{0}(T̂^{Γ}/Z(Ĝ)^{Γ}) satisfying E_{(a,κ′)} ≅ E, all embeddings η, η′ : T̃ ↪ Ĝ such that E_{(a,κ,η)} = E_{(a,κ′,η′)} = (s, ρ), and all representatives s̃ ∈ Ĝ^{ad} of s, the difference κ(s̃, η′) − κ(s̃, η) ∈ Z(Ĝ^{ad})^{Γ} is orthogonal to inv(G′, G) ∈ H^{1}(E, G^{ad}).

Remark A.2. a) Every (E, a, \overline{κ})-admissible inner twisting is E-admissible.

b) If a(T) ⊂ G is elliptic, then every E-admissible inner twisting is (E, a, \overline{κ})- admissible.

c) φ : G → G′ is (E, a_{1}, \overline{κ}_{1})-admissible if and only if ⟨\frac{a′_{1},a_{1};κ_{1}}{a′_{2},a_{2};κ_{2}} ⟩ = ⟨\frac{a′_{1},a_{1};κ′_{1}}{a′_{2},a_{2};κ_{2}} ⟩ for all

representatives κ′_{1} ∈ T̂_{1}^{Γ} of \overline{κ}_{1} ∈ π_{0}(T̂_{1}^{Γ}/Z(Ĝ)^{Γ}) satisfying E_{(a_{1},κ′_{1})} ≅ E.

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Institute of Mathematics, Hebrew University, Givat-Ram, Jerusalem, 91904 Israel

E-mail address: kazhdan@math.huji.ac.il

Institute of Mathematics, Hebrew University, Givat-Ram, Jerusalem, 91904 Israel

E-mail address: vyakov@math.huji.ac.il