T. Kobayashi,

*Multiplicity free theorem in branching problems of unitary
highest weight modules*,

Proceedings of Representation Theory held at Saga, Kyushu, 1997 (K. Mimachi, ed.), 1997, pp. 9-17.

Proceedings of Representation Theory held at Saga, Kyushu, 1997 (K. Mimachi, ed.), 1997, pp. 9-17.

Let π be a unitary highest weight module of a reductive Lie groupG, and (G,G') a reductive symmetric pair such thatG' \hookrightrarrowGinduces a holomorphic embedding of Hermitian symmetric spacesG'/K' \hookrightrarrowG/K. This paper proves that the multiplicity of irreducible representations ofG' occurring in the restriction π|_{G'}is uniformly bounded. Furthermore, we prove that the multiplicity is free if π has a one dimensional minimalK-type. Our method here also establishes an analogous result for the tensor product of unitary highest weight modules, and also for nite dimensional representations of compact groups. Finally, we give an explicit branching formula of a holomorphic discrete series representation π with respect to a semisimple symmetric pair (G,G'). This formula is a generalization of the Kostant-Schmid branching formula which deals with the caseG' =K.

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© Toshiyuki Kobayashi