Hecke algebras for ${\mathrm{G}\mathrm{L}}_n$ over local fields

We study the local Hecke algebra ${\mathcal{H}}_G(K)$ for $G={\mathrm{G}\mathrm{L}}_n$ and $K$ a non-archimedean local field of characteristic zero. We show that for $G={\mathrm{G}\mathrm{L}}_2$ and any two such fields $K$ and $L$, there is a Morita equivalence ${\mathcal{H}}_G(K){\sim }_M{\mathcal{H}}_G(L)$, by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for $G={\mathrm{G}\mathrm{L}}_n$, there is an algebra isomorphism ${\mathcal{H}}_G(K)\cong {\mathcal{H}}_G(L)$ which is an isometry for the induced $L^1$-norm if and only if there is a field isomorphism $K\cong L$.