The Restriction of Unitary Highest Representations of Reductive Lie Groups and its Multiplicities,
International Conference, Summer Solstice Days, Universite de Paris VI & VII, France, June 1999.

Let π be a unitary highest weight module of a semisimple Lie group G, and (G,G') a semisimple symmetric pair. I will talk about theorems:

1. If π has a one dimensional minimal K-type, then the restriction π|_G' is multiplicity free as a G'-module.
2. If the natural map G'/K' \hookrightarrow G/K is holomorphic between Hermitian symmetric spaces, then the restriction π|_G' decomposes discretely into irreducible representations of G' with multiplicity uniformly bounded.

The main results present a number of new settings of multiplicity-free branching laws and also give a unified explanation of various known multiplicity-free results such as the Kostant-Schmid K-type formula of a holomorphic discrete series representation and the Plancherel formula for a line bundle over a Hermitian symmetric space.

I also would like to discuss that such estimate is not true in general for the restriction of non-highest weight modules.

Our approach also yields an analogous result for the tensor product of unitary highest weight modules, and for finite dimensional representations of compact groups.

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