Criteria for discrete decomposability and admissibility in the branching problem

Abstract

In this paper, we study when the restriction of an irreducible representation of a reductive Lie group $G$ to a reductive subgroup $H$ is discretely decomposable. T.\ Kobayashi gave a necessary condition for discrete decomposability in terms of associated varieties. In a previous paper, we proved that the condition is also sufficient if $(G, H)$ is a symmetric pair. The purpose of this paper is to prove the sufficiency for any reductive subgroup $H$ under some hypothesis on $G$ (e.g., $G$ is linear). As an application, we show that discrete decomposability is equivalent to $\mathfrak{h}$-admissibility when the compact factor of $\mathfrak{h}$ is maximal in some sense.