The dual topology for the principal and discrete series on semisimple groups

Abstract:

For a locally compact group $G$, the dual space $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $\hat G$ in the case that $G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $\hat G$; (2) the “discrete series” of square-integrable representations does in fact inherit the discrete topology from $\hat G$; (3) the topology of the reduced dual ${\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.