Multiplicity-free theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs

The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh–Gordan–Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua–Kostant–Schmid K-type formula, and the canonical representations in the sense of Vershik–Gelfand– Graev. Our method works in a uniform manner for both finite and infinite dimensional cases, for both discrete and continuous spectra, and for both classical and exceptional cases.