Title: 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 $G'$ 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.