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

TL;DR

The paper addresses the problem of characterizing the holomorphic discrete series of $SU(1,1)$ through explicit orthogonality relations of matrix elements, closed-form character formulas, and detailed tensor-product multiplicities. It realizes the holomorphic discrete series on Fock-Bargmann spaces $\mathcal{FB}_{\eta}$ with $\eta>\tfrac{1}{2}$, derives Jacobi-polynomial expressions for matrix elements, and establishes the orthogonality relations with the Haar measure. A Clebsch-Gordan-type multiplicity rule is obtained from character considerations on the maximal compact subgroup, yielding $\mathfrak{m}(1,2,3)=\delta_{\eta_3,\eta_1+\eta_2+n}$ for $n\in\mathbb{N}$, extending classical results (Repka). The results provide a solid mathematical foundation for $SU(1,1)$ symmetry in quantum optics, curved spacetime contexts, and related representation-theoretic applications.

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.