A problem on Hecke algebras for $\mathrm{GL}_n(F)$ for $n>2$ over $p$-adic field $F$

Abstract

We study the Hecke algebra $\mathcal{H}_G(F)$ for $G = \mathrm{GL}_n$ and $n>2$ where $F$ is a non-Archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_n$ and $n>2$ and any two such fields $E$ and $F$, there is a Morita equivalence $\mathcal{H}_G(E) \sim \mathcal{H}_G(F)$, by using the Bernstein decomposition of the Hecke algebra and by determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence.