A Hecke algebra isomorphism over close local fields

Abstract

Let $G$ be a split connected reductive group over $\mathbb{Z}$. Let $F$ be a non-archimedean local field. With $K_m:=Ker(G({\mathfrak{O}}_F)\to G({\mathfrak{O}}_F/{\mathfrak{p}}_F^m))$, Kazhdan proved that for a field $F^{\prime }$sufficiently close local field to $F$, the Hecke algebras $\mathcal{H}(G(F),K_m)$ and $\mathcal{H}(G(F^{\prime }),K_m^{\prime })$ are isomorphic, where $K_m^{\prime }$ denotes the corresponding object over $F^{\prime }$. In this article, we generalize this result to general connected reductive groups.

An erratum to rectify the proof of Lemma 2.5 is included