Regular Representations of Time-Frequency Groups

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 G = ⟨ Tk,Ml : k ∈ Z, l ∈ BZ ⟩ , where Tk, Ml are translations and modulations operators acting in L(R), 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 L(R) 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 d = 1, B = 1/L,L ∈ Z and L > 1. 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 d = 1. 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.