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 a concept of “fences” describing interleaving patterns, 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