Stability of Branching Laws for Highest Weight Modules

Abstract

We say a representation V of a group G has stability if its multiplicities m^{G}_{V}(\lambda) is dependent only on some equivalence class of \lambda for a sufficiently large parameter \lambda. In this paper, we prove that the restriction of a holomorphic discrete series representation with respect to any holomorphic symmetric pairs has stability. As a corollary, we give a necessary and sufficient condition on multiplicity-freeness of the branching laws in this setting. This condition is same as the sufficient condition given by the theory of visible actions. We prove a general theorem before we show the stability of holomorphic discrete series representations. Using the general theorem, we also show the stability on quasi-affine spherical homogeneous spaces and the stability of K-type of unitary highest weight modules. We also show that two branching laws of a holomorphic discrete series representation coincide if two subgroups are in same \epsilon-family.