Description
In this talk, we discuss a conceptual perspective on when and how branching multiplicities change in restrictions of representations. We develop a framework for the stability of branching multiplicities arising in the restriction of finite- and infinite-dimensional representations of real reductive Lie groups.
Focusing on pairs whose complexifications are of type (gl(n+1), gl(n)) and (o(n+1),o(n)), we show that branching multiplicities are locally constant on explicitly described convex regions in the joint parameter space of infinitesimal characters, and can change only upon crossing certain hyperplanes (``fences'').
To illustrate the structure concretely, we describe these regions explicitly in the case of tensor products for sl(2), and in connection with this, Pevzner’s talk will discuss the parameter dependence of the blow-up of branching multiplicities in relation to the parameter dependence of Jacobi polynomials.