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

We study the local Hecke algebra $\mathcal{H}_{G}(K)$ for $G = \mathrm{GL}_n$ and $K$ a non-archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_2$ 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 $G = \mathrm{GL}_n$, 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 $K \cong L$.