Level Zero Types and Hecke Algebras for Local Central Simple Algebras

Abstract

Let D be a central division algebra and Ax = GLm(D) the unit group of a central simple algebra over a p-adic field F. The purpose of this paper is to give types (in the sense of Bushnell and Kutzko) for all level zero Bernstein components of Ax and to establish that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras. The types which we consider are lifted from cuspidal representations \tau of M(kD), where M is a standard Levi subgroup of GLm and kD is the residual field of D. Two types are equivalent if and only if the corresponding pairs (M(kD),\tau) are conjugate with respect to Ax. The results are basically the same as in the split case Ax = GLn(F) due to Bushnell and Kutzko. In the non split case there are more equivalent types and the proofs are technically more complicated.