Unitary representations of the universal cover of SU(1,1) and tensor products

TL;DR

This paper explores the connection between weight modules of Lie algebras and unitary representations of the universal cover of SU(1,1), providing new insights into tensor products and the structure of discrete spectra.

Contribution

It introduces new results on the discrete part of tensor products of irreducible representations for the universal cover of SU(1,1).

Findings

New results on the discrete spectrum of tensor products.

Identification of trivial intersections of smooth vectors with discrete spectrum representations.

Enhanced understanding of unitary representations related to Lie algebra modules.

Abstract

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.