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

Relèvement d'intégrales orbitales (transfert) et isomorphismes d'algèbres de Hecke à la Roche

Bertrand Lemaire (1, 2) , Manish Mishra (3)

1 AMU - Aix Marseille Université
2 I2M - Institut de Mathématiques de Marseille
3 University of Pune - India

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 $G'$ of $G$, and there is an injective morphism of ${\Bbb C}$-algebras $\mathcal{H}(G(F),\rho) \rightarrow \mathcal{H}(G'(F),1_{\mathcal{I}'})$ where $\mathcal{H}(G(F),\rho)$ is the Hecke algebra of compactly supported $\rho^{-1}$-spherical functions on $G(F)$ and $\mathcal{I}'$ is an Iwahori subgroup of $G'(F)$. This morphism restricts to an injective morphism $\zeta: \mathcal{Z}(G(F),\rho)\rightarrow \mathcal{Z}(G'(F),1_{\mathcal{I}'})$ 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 ${\rm char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

Domaines: Théorie des représentations [math.RT]

Bertrand Lemaire, Manish Mishra. Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms. Compositio Mathematica, In press, 156 (3), pp.533-603. ⟨10.1112/S0010437X19007838⟩. ⟨hal-02402274⟩