We consider the restriction of irreducible representations of a real reductive group G to a subgroup H. In general, the multiplicity in the branching law can be infinite, even when H is a maximal reductive subgroup, such as in the case where $(G,H)=(GL(p+q), GL(p) ×GL(q))$. In this talk, I plan to start with general results on multiplicities and present criteria for uniform boundedness. Then, we introduce the concept of “fences” for the interlacing pattern, which refine the usual notion of “walls” for Weyl chambers. A theorem will be presented, stating that the multiplicity remains constant unless these “fences” are crossed. If time permits, I will mention some applications to new branching laws.
© Toshiyuki Kobayashi