Title: Hecke algebras for $${{\rm GL}_n}$$ GL n over local fields
Abstract: We study the local Hecke algebra $${\mathcal{H}_{G}(K)}$$ for $${G = {\rm GL}_{n}}$$ and K a non-archimedean local field of characteristic zero. We show that for $${G = {\rm GL}_{2}}$$ 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 $${G = {\rm GL}_{n}}$$ , 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 $${K \cong L}$$ .