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.
Categories: math.RT