Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms

Abstract

Let F be a non-Archimedean local field, G a connected reductive group defined and split over F, and T a maximal F-split torus in G. Let chi(0) be a depth-zero character of the maximal compact subgroup T of T(F). This gives by inflation a character rho of an Iwahori subgroup J subset of T of G(F). From Roche [Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. Ec. Norm. Super. (4) 31 (1998), 361-413], chi(0) defines a reductive F-split group (G) over tilde' whose connected component G' is an endoscopic group of G, and there is an isomorphism of C-algebras H(G(F), rho) -> H((G) over tilde'(F), 1(J)) where H(G(F), rho) is the Hecke algebra of compactly supported p(-1) spherical functions on G(F) and J' is an Iwahori subgroup of G'(F). This isomorphism gives by restriction an injective morphism zeta : Z(G(F), rho) -> Z(G' (F), 1(J')) between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to zeta realizes the transfer (matching of strongly G-regular semi-simple orbital integrals). If char(F) - p > 0, our result is unconditional only if p is large enough.