A Hecke algebra isomorphism over close local fields

TL;DR

This paper generalizes Kazhdan's isomorphism of Hecke algebras over close local fields from split groups to all connected reductive groups, broadening the scope of the original result.

Contribution

It extends Kazhdan's Hecke algebra isomorphism from split groups to arbitrary connected reductive groups over close local fields.

Findings

Hecke algebras over close local fields are isomorphic for general connected reductive groups.

The generalization applies to a wider class of groups beyond split cases.

Supports broader applications in representation theory and number theory.

Abstract

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