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 $(\mathfrak{gl}(n+1), \mathfrak{gl}(n))$ and $(\mathfrak{o}(n+1),\mathfrak{o}(n))$, 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 $\mathfrak{sl}(2)$, 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.
© Toshiyuki Kobayashi