Regular representations of time‐frequency groups

Abstract

In this paper, we study the Plancherel measure of a class of non‐connected nilpotent groups which is of special interest in Gabor theory. Let G be a time‐frequency group. That is where , are translations and modulations operators acting in and B is a non‐singular matrix. We compute the Plancherel measure of the left regular representation of G which is denoted by L. The action of G on induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of L into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut Führ's results which are only obtained for the restricted case where , and Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when . Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.