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

Abstract

Let $F$ be a non-Archimedan 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 $\mathcal{T}$ of $T(F)$. It gives by inflation a character $\rho$ of an Iwahori subgroup $\mathcal{I}$ of $G(F)$ containing $\mathcal{T}$. From Roche, ${\chi }_0$ defines a split endoscopic group $G^{\prime }$ of $G$, and there is an injective morphism of $\mathbb{C}$-algebras $\mathcal{H}(G(F),\rho )\to \mathcal{H}(G^{\prime }(F),1_{{\mathcal{I}}^{\prime }})$ where $\mathcal{H}(G(F),\rho )$ is the Hecke algebra of compactly supported ${\rho }^{-1}$-spherical functions on $G(F)$ and ${\mathcal{I}}^{\prime }$ is an Iwahori subgroup of $G^{\prime }(F)$. This morphism restricts to an injective morphism $\zeta :\mathcal{Z}(G(F),\rho )\to \mathcal{Z}(G^{\prime }(F),1_{{\mathcal{I}}^{\prime }})$ 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 semisimple orbital integrals). If $\mathrm{char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

arXiv comment82 pages

References in corpus (2)

• The base change fundamental lemma for central elements in parahoric Hecke algebras
• On endoscopic transfer of Deligne-Lusztig functions

Cited by in corpus (1)

• Rigid inner forms over local function fields