Unitary highest weight modules
type: Item
instance of: scholarly article
title: Unitary highest weight modules
zbMATH Open document ID: 0623.22014
DOI: 10.1016/0001-8708(87)90049-1
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let G be one of the groups \(SU(p,q)\), \(SO^*(2n)\), or \(\tilde Sp(2n,{\mathbb{R}})\). The problem is to associate the unitary highest weight modules of G with the coadjoint orbits of G and, hence, tie in with the usual picture of the unitary dual of G. Some standard notation: let K be a maximal compact subgroup of G, let \({\mathfrak G}\) and \({\mathfrak K}\) be the complexified Lie algebras of G and K, respectively, and let \(\theta\) be the Cartan involution. Denote by \({\mathfrak l}+{\mathfrak u}^ a \theta\)-stable parabolic subalgebra of \({\mathfrak G}\) and let L be the stabilizer of \({\mathfrak l}\) in G. Finally, let \(\Pi\) be an irreducible unitary representation of L. Considering \(\Pi\) as a representaion of \({\mathfrak l}\) we can write \(\Pi =(\lambda,\pi)\) where \(\lambda\) is in the dual of the center of \({\mathfrak l}\) and \(\pi =\Pi |_{[{\mathfrak l,l}]} \). Using the data \(\lambda,\pi\), the author defines a module (a derived functor module) \(A(\lambda,\pi)\). The association between \(A(\lambda,\pi)\) and a coadjoint orbit of G is given for the cases where \(\pi\) is trivial or \(\pi\) is a unipotent representation of L; according to the main theorem, these are the only cases needed. Theorem. Under certain stated conditions on \(\lambda\) and \(\pi\), \(A(\lambda,\pi)\) is an irreducible unitary holomorphic representation of G. Any irreducible unitary holomorphic representation of G is equivalent to some \(A(\lambda,\pi)\) with \(\lambda,\pi\) satisfying the conditions. Note: The \(A(\lambda,\pi)\) are not all distinct.
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4010813
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: unitary highest weight modules
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: unitary dual
zbMATH Keywords: complexified Lie algebras
zbMATH Keywords: parabolic subalgebra
zbMATH Keywords: irreducible unitary representation
zbMATH Keywords: derived functor module
zbMATH Keywords: unitary holomorphic representation
author: Jeffrey Adams
reviewed by: Thomas A. Farmer
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90049-1
OpenAlex ID: W2031182206
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description: scientific article; zbMATH DE number 4010813