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

We study the Hecke algebra ${\mathcal{H}}_G(F)$ for $G={\mathrm{G}\mathrm{L}}_n$ and $n>2$ where $F$ is a non-Archimedean local field of characteristic zero. We show that for $G={\mathrm{G}\mathrm{L}}_n$ 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.