ENDOSCOPY FOR HECKE ALGEBRAS

Robert E. Kottwitz

1. Introduction. Let G be a p-adic group and let f ∈ C_{c}^{∞}(G). Let H be an endoscopic group for G. The conjectural theory of endoscopy predicts th e existence of a function f^{H} ∈ C_{c}^{∞}(H) whose stable orbital integrals are linear comb inations of orbital integrals of f, the coefficients in the linear combinations being the transfe r factors of Langlands-Shelstad [LS]. For a general function f one does not expect to be able to find f^{H} explicitly. But if G is unramified and f is bi-invariant under a hype rspecial maximal compact subgroup K of G, then at least there is a conjectural candidate fo r f^{H}. This conjecture, which is due to Langlands, is known as the fundamental lemma. Recen tly Waldspurger [W] has shown that the fundamental lemma would imply the existence of f^{H} for general functions f.

It would be desirable to have a greater supply of functions f for which one k nows a conjectural candidate for f^{H}. For example, Assem, in his work on endoscopic induction of unipotent orbital integrals, was obliged to limit his test functions to those bi-invariant under a hyperspecial maximal compact subgroup. However these are not enough to separate the various unipotent orbits and are thus not completely satisfactory for the purpose he had in mind, which was to test the correctness of his ideas on transfer factors for unipotent elements (the transf er factors of Langlands-Shelstad are defined only for certain semisimple elements, including all regular ones). Motivated by discussions with Assem during his visit to Chicago in the spring of 1996, I looked for more candidate pairs f, f^{H} and found the ones that are the subject of this lecture.

2. Preliminaries. Let F be a p-adic field and let \overline{F} be an algebraic closure of F. Let 𝔬 be the valuation ring of F. Let G be a split adjoint semisimple group over F. We write Ĝ for the Langlands dual group of G. Sometimes we write G rather tha n G(F).

3. Stable conjugacy. Recall that a semisimple element γ ∈ G(F) is said to be strongly regular if its centralizer G_{γ} in G is a torus. Let γ, γ′ be strongly regular semisimple elements in G(F). Recall that γ, γ′ are said to be stably conjugate if there exists g ∈ G(\overline{F}) such that g^{−1}γg = γ′. The inner automorphism x ↦ g^{−1}xg induces a n isomorphism (over F) from G_{γ} to G_{γ′}, and this isomorphism is independent of t he choice of g.

4. Stable orbital integrals. Let γ be a strongly regular sem isimple element of G(F). Choose Haar measures dg and dt on G(F) and G_{γ}(F) respecti vely. For any function f ∈ C_{c}^{∞}(G) we write O_{γ}(f) for the orbital integral

O_{γ}(f) := ∫_{G(F)/G_{γ}(F)} f(gγg^{−1}) dg/dt.

Typeset by AMS-TEX

Similarly we write SO_{γ}(f) for the stable orbital integral

SO_{γ}(f) := ∑_{γ′} O_{γ′}(f)

where the sum is taken over a set of representatives γ′ for the conjugac y classes within the stable conjugacy class of γ. We form the orbital integral O_{γ′} using dg and the Haar measure dt′ on G_{γ′}(F) obtained from dt via the isomorphism from G_{γ} to G_{γ′} defined in §3.

5. Endoscopic groups. Let s be a semisimple element in Ĝ a nd let Ĥ denote its centralizer, a reductive complex group which is in fact connected by a theorem of Steinberg (since G is adjoint, Ĝ is simply connected). Let H be the split connected reductive group over F whose dual group is Ĥ; H is called an endoscopic group for G. Note that H comes equipped with the element s ∈ Z(Ĥ), where Z(Ĥ) denote s the center of Ĥ.

6. Transfer factors. Let T_{G} denote a split maximal torus in G and let W_{G} denote its Weyl group. The quotient variety T_{G}/W_{G} is the space of semisimple conju gacy classes in G, and its set of F -points is the set of semisimple conjugacy classes that are d efined over F. The same remarks apply to H; moreover there is an isomorphism T_{G} ≃ T_{H}, c anonical up to composition with elements in W_{G}, and via this isomorphism W_{H} goes over to a subgroup of W_{G}. Therefore there is a canonical morphism from T_{H}/W_{H} to T_{G}/W_{G} and hence a canonical map from the set of semisimple conjugacy classe s in H over F to the set of semisimple conjugacy classes in G over F. One says that a semis imple conjugacy class in H is strongly G-regular if the associated class in G is strongly regul ar.

Let Δ_{0} be the transfer factor defined just before Theorem 3.7.A of [LS]. It depends on the choice of an F -splitting for G, but since G is adjoint any two F - sp littings are G(F)-conjugate and hence the transfer factor is independent of the choice of F -splitting. Since G and H are split there is a canonical L-homomorphism LH → ^{L}G, n amely the obvious inclusion, and of course it is this L-homomorphism we use to form the transfe r factor Δ_{0}.

7. Matching orbital integrals. We now recall another definition fro m [LS]. Let f ∈ C_{c}^{∞}(G) and let f^{H} ∈ C_{c}^{∞}(H). One says that f and f^{H} have matching orbital integrals if for every strongly G-regular element γ_{H} ∈ H(F) the following equality holds:

SO_{γ_{H}} (f^{H}) = ∑_{γ′} Δ_{0}(γ_{H}, γ′)O_{γ′}(f),

where γ′ runs over a set of representatives for the G(F)-conjugacy cla sses of elements in the stable conjugacy class in G(F) associated to the stable conjugacy class o f γ_{H} in H(F) as in §6. Langlands and Shelstad conjecture that for every f ∈ C^{∈}fty_{c}(G) there exists f^{H} ∈ C_{c}^{∞}(H) such that f and f^{H} have matching orbital in tegrals.

8. Statement of fundamental lemma. We now recall from [L] the statement of the (conjectural) fundamental lemma. Fix special maximal compact subgroups K_{G} an d K_{H} in G(F) and H(F) respectively. The Satake isomorphism identifies the Hecke algebra C_{c}(K_{G}\G(F)/K_{G}) with the algebra of Ĝ-conjugation invariant regular functions on Ĝ, and the same is true for H. By means of these identifications the obvious restriction map from functions on Ĝ to functions on Ĥ determines an algebra homomorphism

b : C_{c}(K_{G}\G(F)/K_{G}) → C_{c}(K_{H}\H(F)/K_{H}).

The fundamental lemma is the assertion that for any f ∈ C_{c}(K_{G}\G(F)/K_{G}) the functions f and b(f) have matching orbital integrals.

9. Deligne-Lusztig representations. We now review a construction fr om [DL]. Let k be the residue field of F. We now consider the group G over the field k. As before we denote by W_{G} the Weyl group of G. Let X denote the flag manifold of G, a variety over k. The variety X is a d isjoint union of locally closed pieces X_{w}, one for each element w ∈ W_{G}. By definition th e subset X_{w} consists of all points x ∈ X such that x and F (x) are in relative posit ion w (in other words, the pair (x, F (x)) lies in the G-orbit in X ×X indexed by w) ; here F denotes the Frobenius morphism X → X. As in Definition 1.5 of [DL] we denote b y R_{w} the virtual character

∑(−1)^{i}H_{c}^{i}(X_{w}, \overline{Q}_{l})

of G(k) (here l is a prime different from the characteristic of k). The v irtual character R_{w} depends only on the conjugacy class of w ∈ W_{G} (see Theorem 1.6 in [DL]) and its values are rational integers i ndependent of l (see Proposition 3.3 of [DL]). As in [Lu] we define, for any c omplex-valued class function ξ on W_{G}, a complex-valued class function R_{ξ} on G(k) by

R_{ξ} := |W_{G}|^{−1} ∑_{w∈W_{G}} ξ(w)R_{w}

(in this definition we are of course regarding R_{w} as a Z-valued func tion on G(k)).

10. Some test functions. We now use the class functions R_{ξ} to define functions f_{ξ} on G(F). T o do this we consider the canonical surjection

K_{G} = G(𝔬) → G(k),

which we denote by x ↦ x̄. We then def ine f_{ξ} on K_{G} by putting

f_{ξ}(x) = R_{ξ}(x̄)/ vol(K_{G})

and extend f_{ξ} to G(F) by making it 0 outside of K_{G}.

11. Conjecture. As before we use the natural embedding of W_{H} as a subgroup of W_{G}. This embedding is canonical up to inner automorphisms of W_{G}, and ther efore we can restrict a class function ξ_{G} on W_{G} to W_{H}, obtaining a class functio n ξ_{H} on W_{H}. It seems very likely that the functions f_{ξ_{G}} ∈ C_{c}^{∞}(G) and f _{xi_{H}} ∈ C_{c}^{∞}(H) have matching orbital integrals.

References

[DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161.

[L] R. P. Langlands, Les débuts d’une formule des traces sta ble, Publ. Math. Univ. Paris VII, vol. 13, Paris, 1983.

[LS] R. P. Langlands and D. Shelstad, On the definition of tr ansfer factors, Math. Ann. 278 (1987), 219–271.

[Lu] G. Lusztig, Characters of Reductive Groups over a Fin ite Field, Ann. of Math. Studies, 107, Princeton Univ. Press, 1984.

[W] J.-L. Waldspurger, Le lemme fondamental implique le trans fert, Compositio Math. 105 (1997), 153–236.