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

• Published in: Compositio mathematica Vol. 156; no. 3; pp. 533 - 603
• Main Authors: Lemaire, Bertrand, Mishra, Manish
• Format: Journal Article
• Language: English
• Published: London Cambridge University Press 01.03.2020; Foundation Compositio Mathematica
• Subjects: Algebra; Fields (mathematics); Integrals; Isomorphism; Matching; Mathematics; Representation Theory; Set theory; Subgroups; Toruses
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1112/S0010437X19007838

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.