Regular representations of time-frequency groups (Q2922205)

This paper focuses on the Plancherel measure of a class of non-connected nilpotent groups. Let \(G=\langle T_k,\,M_l:\;k\in\mathbb{Z}^d,\;l\in B\mathbb{Z}^d\rangle\) be a time-frequency group, where \(T_k\) and \(M_l\) are translation and modulation operators on \(L^2(\mathbb{R}^d)\) and \(B\) is a non-singular matrix. The authors compute the decomposition of the Plancherel measure \(L\) of the left regular representation of \(G\) into a direct integral of irreducible representations, via providing a precise description of the unitary dual and its Plancherel measure. As an application, a direct integral decomposition of the Gabor representation of \(G\) with an integral matrix \(B\) is obtained.