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:
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.
- 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.
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