标题： SU(1,1)的全纯离散列：正交关系、特征公式以及张量积分解中的重数 显示英文标题

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

摘要： SU$(1,1)$群在物理学的各个领域中扮演着基础角色，包括量子力学、量子光学和表示论。 在这项工作中，我们重新审视了 SU$(1,1)$的全纯离散系列表示，重点关注矩阵元的正交关系、酉不可约表示（UIR）的特征公式以及这些 UIR 的张量积分解。 特别注意了这些分解的结构及其相关的多重性，这对于理解 SU$(1,1)$对称框架内的复合系统和相互作用至关重要。 这些发现为 SU$(1,1)$表示的数学基础及其在理论物理中的意义提供了更深入的见解。 显示英文摘要

摘要： 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.