*The Restriction of Unitary Highest Representations of Reductive Lie
Groups and its Multiplicities*,

International Conference, Summer Solstice
Days, Université 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:
- If π has a one dimensional minimal
*K*-type, then the restriction π|_{G'} is multiplicity free as a *G*'-module.
- 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.

© Toshiyuki Kobayashi