Representations of exponential solvable Lie groups induced from maximal subgroups
type: Item
instance of: scholarly article
title: Representations of exponential solvable Lie groups induced from maximal subgroups
zbMATH Open document ID: 0801.22004
author: Ronald L. Lipsman
published in: Michigan Mathematical Journal
publication date: 1994-04-24T00:00:00Z
review text: Let \(G\) be an exponential solvable Lie group. The author studies in detail the mechanism of inducing process up to \(G\) from a unitary representation of a maximal closed connected proper subgroup \(H\). Namely, let \(\nu\) be an irreducible unitary representation of \(H\) and let \(\pi = \text{Ind}^ G_ H \nu\) be the unitary representation of \(G\) induced from \(\nu\). The author analyses this situation in the framework of the orbit method. When \(\dim G/H = 1\), this analysis was previously given by himself [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 127-164 (1990; Zbl 0724.22006)]. This paper treats the case where \(\dim G/H = 2\) and derives seven mutually exclusive possibilities for the structure of the inducing up. Let \(\mathfrak g\) (resp. \(\mathfrak h\)) be the Lie algebra of \(G\) (resp. \(H\)) and let \(\Omega_ H(\nu) \subset {\mathfrak h}^*\) be the coadjoint \(H\)- orbit corresponding to \(\nu\). There is the unique 1-codimensional subalgebra \(\mathfrak k\) of \(\mathfrak h\) which is a 3-codimensional ideal in \(\mathfrak g\). One takes \(\psi\) in \(\Omega_ H(\nu)\) and considers the irreducible unitary representation \(\gamma\) of \(K = \text{exp }{\mathfrak k}\), which corresponds to \(\theta = \psi| {\mathfrak k}\). Then the key determining factor is the stability subgroup \(G(\gamma)\) of \(\gamma\) in \(G\).
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 474972
zbMATH Keywords: induced representations
zbMATH Keywords: exponential solvable Lie group
zbMATH Keywords: unitary representation
zbMATH Keywords: orbit method
zbMATH Keywords: Lie algebra
zbMATH Keywords: stability subgroup
Wikidata QID: Q115238978
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004754
OpenAlex ID: W2021756474
DOI: 10.1307/MMJ/1029004754
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description: scientific article; zbMATH DE number 474972