2013/07/02 by Masatoshi Kitagawa, Kitagawa, Masatoshi
Mathematics · #20G05 #22E46 #32M15 #57S20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1307.0606
openalex publication_date 2013/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say a representation V of a group G has stability if its multiplicities mGV(λ) is dependent only on some equivalence class of λfor a sufficiently large parameter λ. 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 ε-family.