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Coadjoint geometry for discretely decomposable restrictions of certain series of representations of indefinite unitary groups

Salma Nasrin

Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

https://doi.org/10.1142/S0129167X17500744Cited by:1 (Source: Crossref)

Abstract

Zuckerman’s derived functor module of a semisimple Lie group $G$ yields a unitary representation $π$ which may be regarded as a geometric quantization of an elliptic orbit $𝒪$ in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations $π$ of the indefinite unitary group $G = U (p, q)$ and a family of subgroups $H$ of $G$ such that the restriction $π |_{H}$ is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of $A_{q} (λ)$ with respect to reductive subgroups, II, Ann. of Math.147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci.41 (2005) 497–549), where $π$ is not necessarily tempered and $H$ is not necessarily compact. We prove that the corresponding moment map $μ : 𝒪 \to 𝔥^{*}$ is proper, determine the image $μ (𝒪)$ , and compute the Corwin–Greenleaf multiplicity function explicitly.

Keywords:

AMSC: 22E46, 22E60, 32M15, 53C35, 81S10

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