Let π be a unitary highest weight module of a reductive Lie group G, and (G, G') a reductive symmetric pair such that G' \hookrightrarrow G induces a holomorphic embedding of Hermitian symmetric spaces G'/K' \hookrightrarrow G/K. This paper proves that the multiplicity of irreducible representations of G' occurring in the restriction π|G' is uniformly bounded. Furthermore, we prove that the multiplicity is free if π has a one dimensional minimal K-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 case G' = K.
The original publication is available here (pdf).
© Toshiyuki Kobayashi