Weak containment and tensor products of group representations. II

Abstract

Let G be a locally compact group and G the set of equivalence classes of irreducible unitary representations of G. (~, equipped with the Jacobson topology, is called the dual space of G. In this paper, we continue our study of problems related to supports in G of tensor products of representations of G. It is well known that ifn and ~ are finite dimensional representations of G and ~g denotes the conjugate representation of Q, then the tensor product n| contains the trivial one-dimensional representation 1G if and only ifg and Q have a common subrepresentation. Can this be generalized to the case, where n and ~ are arbitrary and containment is replaced by weak containment? This question was first discussed by Fell [6] and has been taken up in [4] and [21]. The object of Sect. 1 is to add some further results concerning this problem. Our main results are the following. We show that, for a polynomially growing discrete group G and any two unitary representations n and ~ of G, 1, is weakly contained in nQff (i.e. 1~ ~suppn| ) if and only if suppnnsupp04:0. This extends the corresponding result for locally nilpotent discrete groups in [21]. If G is a motion group and n e (~, we give a criterion on n to satisfy that laesuppn| ~ is equivalent to ff=n for all ~Ed.