  In this paper we initiate a study of the relation between weight modules for
simple Lie algebras and unitary representations of the corresponding
simply-connected Lie groups. In particular we consider in detail from this
point of view the universal covering group of SU(1,1), including new results on
the discrete part of tensor products of irreducible representations. As a
consequence of these results, we show that the set of smooth vectors of the
tensor product intersects trivially some of the representations in the discrete
spectrum.
