On unitary representations of algebraic groups over local fields

Abstract

Let G \mathbf {G} be an algebraic group over a local field k \mathbf {k} of characteristic zero. We show that the locally compact group G ( k ) \mathbf {G}(\mathbf {k}) consisting of the k \mathbf {k} -rational points of G \mathbf {G} is of type I. Moreover, we complete Lipsman’s characterization of the groups G \mathbf {G} for which every irreducible unitary representation of G ( k ) \mathbf {G}(\mathbf {k}) is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.