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 $\unicode[STIX]{x1D712}_{0}$ be a depth-zero character of the maximal compact subgroup $T$ of $T(F)$. This gives by inflation a character $\unicode[STIX]{x1D70C}$ of an Iwahori subgroup $\unicode[STIX]{x2110}\subset T$ of $G(F)$. From Roche [Types and Hecke algebras for principal series representations of split reductive$p$-adic groups, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 361–413], $\unicode[STIX]{x1D712}_{0}$ defines a reductive $F$-split group $\widetilde{G}^{\prime }$ whose connected component $G^{\prime }$ is an endoscopic group of $G$, and there is an isomorphism of $\mathbb{C}$-algebras $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ where $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ is the Hecke algebra of compactly supported $\unicode[STIX]{x1D70C}^{-1}$-spherical functions on $G(F)$ and $\unicode[STIX]{x2110}^{\prime }$ is an Iwahori subgroup of $G^{\prime }(F)$. This isomorphism gives by restriction an injective morphism $\unicode[STIX]{x1D701}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $\unicode[STIX]{x1D701}$ realizes the transfer (matching of strongly $G$-regular semi-simple orbital integrals). If $\operatorname{char}(F)=p>0$, our result is unconditional only if $p$ is large enough.