Hecke algebras for GL n over local fields

We study the local Hecke algebra HG(K) for G= GL n and K a non-archimedean local field of characteristic zero. We show that for G= GL 2 and any two such fields K and L, there is a Morita equivalence HG(K) ∼ MHG(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= GL n, there is an algebra isomorphism HG(K) ≅ HG(L) which is an isometry for the induced L1-norm if and only if there is a field isomorphism K≅ L