Coadjoint orbits and reductive dual pairs
type: Item
instance of: scholarly article
title: Coadjoint orbits and reductive dual pairs
zbMATH Open document ID: 0623.22015
DOI: 10.1016/0001-8708(87)90050-8
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Suppose \((\tilde G,\tilde G')\) is a reductive dual pair of subgroups of the metaplectic group \(\tilde Sp(2n,{\mathbb{R}})\) with \(\tilde G\) compact. In an earlier paper (see the preceding review), the author established an association between the unitary highest weight modules of a classical simple Lie group, such as \(\tilde G'\), and the coadjoint orbits - that is, he showed how to quantize the orbits. But it is known about reductive dual pairs that the restriction of the oscillator representation of \(Sp(2n,{\mathbb{R}})\) to \(\tilde G\tilde G'\), decomposes as \(\sum \tau_ i \otimes \tau_ i'\) where the correspondence \(\tau_ i\leftrightarrow \tau_ i'\) is a bijection between the unitary highest weight modules of the two groups. Thus, we have two realizations of the unitary highest weight modules of \(\tilde G'\). How are the two connected ? The main result of this paper is a theorem which describes the bijection \(\tau_ i\leftrightarrow \tau_ i'\) as a correspondence between quantizations of orbits \(\pi({\mathcal O})\leftrightarrow \pi'({\mathcal O}')\).
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4010814
zbMATH Keywords: reductive dual pair
zbMATH Keywords: metaplectic group
zbMATH Keywords: unitary highest weight modules
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: quantizations of orbits
author: Jeffrey Adams
reviewed by: Thomas A. Farmer
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90050-8
OpenAlex ID: W1999274475
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description: scientific article; zbMATH DE number 4010814