Admissible vectors for the regular representation

Abstract

It is well known that for irreducible, square-integrable representations of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admissible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group $G$ with type I regular representation $\lambda_G$, we show that $\lambda_G$ itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff $G$ is nonunimodular.