Holomorphic Discrete Series of SU(1,1): Orthogonality Relations, Character Formulas, and Multiplicities in Tensor Product Decompositions

Abstract: The SU$(1,1)$ group plays a fundamental role in various areas of physics, including quantum mechanics, quantum optics, and representation theory. In this work we revisit the holomorphic discrete series representations of SU$(1,1)$, with a focus on orthogonality relations for matrix elements, character formulas of unitary irreducible representations (UIRs), and the decomposition of tensor products of these UIRs. Special attention is given to the structure of these decompositions and the associated multiplicities, which are essential for understanding composite systems and interactions within SU$(1,1)$ symmetry frameworks. These findings offer deeper insights into the mathematical foundations of SU$(1,1)$ representations and their significance in theoretical physics.