Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs. (English) Zbl 1304.22013

Kobayashi, Toshiyuki

Kobayashi, Toshiyuki (ed.) et al., Representation theory and automorphic forms. Based on the symposium, Seoul, Korea, February 14–17, 2005. Basel: Birkhäuser (ISBN 978-0-8176-4505-2/hbk). Progress in Mathematics 255, 45-109 (2008).

Summary: 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 known results such as the Clebsch-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.
For the entire collection see [Zbl 1124.11004].

Reviewer: Reviewer (Berlin)

MSC:

22E46	Semisimple Lie groups and their representations
05E15	Combinatorial aspects of groups and algebras (MSC2010)
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35	Differential geometry of symmetric spaces

Keywords:

multiplicity-free representation; branching rule; symmetric pair; highest weight module; Hermitian symmetric space; reproducing kernel; semisimple Lie group