Inversion formula for twisted orbital integrals on real reductive Lie groups
type: Item
instance of: scholarly article
title: Inversion formula for twisted orbital integrals on real reductive Lie groups
zbMATH Open document ID: 0883.22008
author: David Renard
published in: Journal of Functional Analysis
publication date: 1997-08-07T00:00:00Z
review text: In this sequel to his article [ibid. 145, 374-454 (1997; Zbl 0877.22002)], the author proves an inversion formula for twisted orbital integrals on a reductive Lie group \(G\). These integrals are obtained by integrating a smooth function \(f\) with compact support on the twisted conjugacy class of \(x\), i.e. the set of all \(gx\theta (g^{-1})\) for \(g\in G\), where \(\theta\) is a finite order automorphism of \(G\). The main result of the paper is a Fourier inversion formula, expressing the orbital integral as a sum of integrals of \(\Theta_\chi (f) \Psi_\chi (x)\) (simplified notation), where the variable \(\chi\) runs over some character group, and the summation runs over the set of conjugacy classes of Cartan subspaces. The \(\theta_\chi\)'s are invariant eigendistributions under the action of biinvariant differential operators, and the \(\Psi_\chi\)'s are specific orbital functions. The construction and asymptotic study of the \(\Psi_\chi\)'s make up the most important part of the present paper.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 1045911
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: twisted orbital integrals
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Fourier inversion formula
zbMATH Keywords: differential operators
reviewed by: François Rouvière
MaRDI profile type: Publication
full work available at URL: jfan.1996.3047
OpenAlex ID: W1966383008
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DOI: 10.1006/JFAN.1996.3047
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description: scientific article; zbMATH DE number 1045911