Unitary highest weight representations via the orbit method. I: The scalar case
type: Item
instance of: scholarly article
title: Unitary highest weight representations via the orbit method. I: The scalar case
zbMATH Open document ID: 0864.22006
DOI: 10.1007/BF00116520
author: Joachim Hilgert
author: Karl-Hermann Neeb
author: Bent Ørsted
published in: Acta Applicandae Mathematicae
publication date: 1997-06-22T00:00:00Z
review text: The purpose of the paper is to describe realizations of unitary highest weight representations of an Hermitian simple Lie group \(G\) by using the orbit method. Let \(Q\) be a parabolic subgroup associated with an \(sl_2\)-triple \((H,X,Y)\) and a subgroup \(Q_{\text{red}}\) of \(Q\). By the geometrization process the authors associate to the adjoint orbit \({\mathcal O}_X\) a unitary representation of \(Q_{\text{red}}\). Since the orbit is of convex type, i.e. is contained in a pointed convex cone, the Laplace transform can be used to realize this representation in a space \({\mathcal H}\) of holomorphic functions whose reproducing kernel is computed. Further this representation is extended to the subgroup \(Q\). Then one sees that the Hilbert space \({\mathcal H}\) is related to a unitary highest weight representation of \(G\) by comparing the reproducing kernels.
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 943745
zbMATH Keywords: Siegel domains
zbMATH Keywords: nilpotent orbits
zbMATH Keywords: highest weight representations
zbMATH Keywords: simple Lie group
zbMATH Keywords: orbit method
zbMATH Keywords: Laplace transform
zbMATH Keywords: Hilbert space
zbMATH Keywords: reproducing kernels
reviewed by: Jacques Faraut
MaRDI profile type: Publication
cites work: Differential operators and highest weight representations
cites work: Q3318972
cites work: An intrinsic analysis of unitarizable highest weight modules
cites work: Function spaces and reproducing kernels on bounded symmetric domains
cites work: Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations
cites work: Q3897665
cites work: Q4693080
cites work: Q4838916
cites work: Q4878268
cites work: Hermitian symmetric spaces and their unitary highest weight modules
cites work: Q4884651
cites work: Limits of holomorphic discrete series
cites work: Coherent states, holomorphic extensions, and highest weight representations
cites work: Holomorphic representation theory. II
cites work: Singular unitary representations and indefinite harmonic theory
cites work: Q3941676
cites work: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group
cites work: Q5682517
full work available at URL: https://doi.org/10.1007/bf00116520
OpenAlex ID: W2063606907
description: scientific article; zbMATH DE number 943745