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 $(G,H)=(\mathit{GL}(p+q,\mathbb{R}),\mathit{GL}(p,\mathbb{R}) \times \mathit{GL}(q,\mathbb{R}))$.In this talk, I plan to start with general results on multiplicities, presenting criteria for their finiteness and uniform boundedness, respectively.
I will then move on to a more detailed analysis of branching in the case of bounded multiplicity, within the context of strong Gelfand pairs. We introduce the concept of ''fences for the interlacing pattern, which refines the usual notion of ''walls for Weyl chambers. A theorem will be presented, stating that the multiplicity remains constant unless these ''fences are crossed. This approach will be illustrated with examples of both tempered and non-tempered representations, as well as a non-vanishing theorem for period integrals of pairs of reductive symmetric spaces.
© Toshiyuki Kobayashi