On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Abstract

Let H be a locally compact group and K be a locally compact abelian group. Also let G=H×τK denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L2(K)) on G defined by \(U(h,x)f(y)=\delta(h)^{\frac{1}{2}}f(\tau_{h^{-1}}(yx^{-1}))\) is called the quasi regular representation. The properties of this representation in the case K=(ℝn,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) \(\delta\Delta_{H}\not\equiv 1\) , and (ii) the stabilizers Hω are compact for a.e. \(\omega \in \widehat{K}\) ; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.