The Poisson- Plancherel formula for a real semialgebraic group. I: Fourier transform of orbital integrals
type: Item
instance of: scholarly article
title: The Poisson- Plancherel formula for a real semialgebraic group. I: Fourier transform of orbital integrals
zbMATH Open document ID: 0836.22021
author: Mohamed Salah Khalgui
author: Pierre Torasso
published in: Journal of Functional Analysis
publication date: 1996-05-06T00:00:00Z
review text: This paper is the first part of the authors' work to derive the Poisson- Plancherel formula for semialgebraic groups \(G\), a formula conjectured by \textit{M. Vergne} [A Plancherel formula without group representations, O. A. G. R. Conference INCREST, Bucarest, Roumania (1980)]. Let \(\mathfrak g\) be the Lie algebra of \(G\). Basing on works of \textit{M. Duflo} [see, for instance, Acta Math. 149, 153-213 (1982; Zbl 0529.22011)], they associate to any linear form on \(\mathfrak g\) a subgroup, called canonical acceptable subgroup, of \(G\) and study the conjugacy classes of the reductive factors of these subgroups. They then define orbital integrals on \(\mathfrak g\), generalizing those introduced by \textit{Harish-Chandra} [Ann. Math. 80, 551-593 (1964; Zbl 0152.134)] and \textit{M. Duflo} [Représentations unitaires des groupes de Lie et méthode des orbites. Actualités mathématiques, Actes 6ième Congr. Group. Math. Exp. Latine, Luxembourg 1981, 125-138 (1982; Zbl 0492.22008)]. The main result of this paper is an expression of the Fourier transforms of these orbital integrals. We consider a connected central subgroup \(Z\) of \(G\) contained in the unipotent radical of \(G\). Let \(\mathfrak z\) be the Lie algebra of \(Z\) and let \(\chi\) be a unitary character of \(Z\). To any elliptic element \(T\) of \(\mathfrak g\), they associate a \(\chi\)-tempered distribution \(M_{G, T, \chi}\), called \(\chi\)-orbital integral, whose Fourier transform is denoted by \(\Theta_{G, T, \chi}\). If we put \({\mathfrak g}_\chi^* = \{\xi \in {\mathfrak g}^*;\) \(\xi \mid {\mathfrak z} = (d \chi /\sqrt {-1}) |{\mathfrak z}\}\), \(\Theta_{G, T, \chi}\) is a tempered generalized function on \({\mathfrak g}^*_\chi\). When \(Z\) is trivial, we write simply \(M_{G, T}\) and \(\Theta_{G, T}\). In this case \(M_{G, T}\) is a tempered distribution on \(\mathfrak g\), supported by the \(G\)-orbit of \(T\). When \(G\) is reductive, \(M_{G, T}\) is, up to a scalar multiplication, the Haris…
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 515262
zbMATH Keywords: Poisson-Plancherel formula
zbMATH Keywords: semialgebraic groups
zbMATH Keywords: Lie algebra
zbMATH Keywords: linear form
zbMATH Keywords: canonical acceptable subgroup
zbMATH Keywords: orbital integrals
zbMATH Keywords: Fourier transforms
zbMATH Keywords: \(\chi\)-tempered distribution
zbMATH Keywords: tempered generalized function
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
full work available at URL: jfan.1993.1117
OpenAlex ID: W2071746615
DOI: 10.1006/JFAN.1993.1117
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description: scientific article; zbMATH DE number 515262