Représentations monomiales des groupes de Lie nilpotents. (Monomial representations of nilpotent Lie groups)
type: Item
instance of: scholarly article
title: Représentations monomiales des groupes de Lie nilpotents. (Monomial representations of nilpotent Lie groups)
zbMATH Open document ID: 0588.22008
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: In this paper, we study by the method of orbits due to \textit{A. A. Kirillov} [Usp. Mat. Nauk 17, No.4 (106), 57-110 (1962; Zbl 0106.250)] monomial representations with finite multiplicities of nilpotent Lie groups. For a unitary representation \(\sigma\) of a Lie group, we denote by \({\mathcal H}_{\sigma}\) its Hilbert space and by \({\mathcal H}_{\sigma}^{\pm \infty}\) the space of \(C^{\pm \infty}\)-vectors for \(\sigma\). Let G be a connected and simply connected nilpotent Lie group, H an analytic subgroup of G and let \(\chi\) be a unitary character of H. We consider the induced representation \(\tau =ind^ G_ H \chi\) of G and its canonical central decomposition \(\tau =\int_{\hat G}^{\oplus} m(\pi)\pi d\nu (\pi)\), \(\hat G\) being the unitary dual of G, with a certain Borel measure \(\nu\) on \(\hat G\) and multiplicity function m: \(\hat G\ni \pi \mapsto m(\pi)\in {\mathbb{Z}}_+\cup \{+\infty \}.\) According to this decomposition, an element \(a_{\tau}\in {\mathcal H}_{\tau}^{-\infty}\) given by \(a_{\tau}: {\mathcal H}_{\tau}^{+\infty}\ni \phi \mapsto \overline{\phi (e)}\in {\mathbb{C}}\) (e : the unit of G) is decomposed into \(a_{\tau}=\int_{\hat G}^{\oplus} (\sum^{m}_{k=1}\oplus a^ k_{\pi}) d\nu (\pi)\) [Penney's abstract Plancherel formula, see \textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016)] with some \(a^ k_{\pi}\in {\mathcal H}_{\pi}^{-\infty}.\) Assuming m(\(\pi)\) finite almost everywhere for \(\nu\), we give these \(a^ k_{\pi}\) explicitly to write down a concrete Plancherel formula for \(\tau\).
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 3944169
zbMATH Keywords: method of orbits
zbMATH Keywords: monomial representations
zbMATH Keywords: finite multiplicities
zbMATH Keywords: nilpotent Lie groups
zbMATH Keywords: unitary representation
zbMATH Keywords: induced representation
zbMATH Keywords: Plancherel formula
author: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.329
OpenAlex ID: W1979545278
DOI: 10.2140/PJM.1987.127.329
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description: scientific article; zbMATH DE number 3944169