On unitary representations of algebraic groups over local fields

Abstract

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