Orbital integrals on reductive Lie algebras
type: Item
instance of: scholarly article
title: Orbital integrals on reductive Lie algebras
zbMATH Open document ID: 0814.22005
DOI: 10.1007/BF01231757
author: Abderrazak Bouaziz
published in: Inventiones Mathematicae
publication date: 1994-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144166
review text: Let \(G\) be a reductive Lie group of Harish-Chandra class, with Lie algebra \(\mathfrak g\). Let \(U\) be a completely \(G\)-invariant open subset of \(\mathfrak g\) (under the adjoint action), and let \({\mathcal D}(U)\) be the space of compactly supported smooth functions on \(U\). To each \(f\) in \({\mathcal D}(U)\) correspond orbital integrals \(Jf(X)\), obtained when integrating \(f\) on the orbits \(G\cdot X\) of regular elements \(X\) in \(U\), by means of the Liouville measure on these orbits. The main result of the paper is a complete description of the image of the map \(J\), as a space of \(G\)-invariant smooth functions on the subset of regular semisimple elements in \(U\), which satisfy certain jump conditions. Among the tools used in its proof are \(G\)-invariant partitions of unity and Harish-Chandra's method of descent: the latter reduces the problem for \(\mathfrak g\) to a similar problem for the centralizer of a semisimple element, thus allowing a proof by induction on the dimension. An analogous result is proved for the Schwartz space \({\mathcal S}({\mathfrak g})\) replacing \({\mathcal D}(U)\) above. Transposing the map \(J\), the author also describes the dual spaces \({\mathcal D}'(U)^ G\) and \({\mathcal S}'({\mathfrak g})^ G\) of \(G\)-invariant distributions. For orbital integrals on the group \(G\) itself, see [the author, Ann. Sci. Ec. Norm. Supér., IV. Sér. 27, 573-609 (1994)], which relies on the results of the present paper.
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 549842
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: adjoint action
zbMATH Keywords: compactly supported smooth functions
zbMATH Keywords: orbital integrals
zbMATH Keywords: regular elements
zbMATH Keywords: \(G\)-invariant smooth functions
zbMATH Keywords: regular semisimple elements
zbMATH Keywords: Schwartz space
zbMATH Keywords: dual spaces
zbMATH Keywords: \(G\)-invariant distributions
reviewed by: François Rouvière
MaRDI profile type: Publication
cites work: The \(L^ 2\)-Lefschetz numbers of Hecke operators
cites work: Sur les caractères des groupes de Lie réductifs non connexes. (On the characters of non-connected reductive Lie groups)
cites work: Q4173774
cites work: ASYMPTOTIC PROPERTIES OF POLYNOMIALS AND ALGEBRAIC FUNCTIONS OF SEVERAL VARIABLES
cites work: Invariant eigendistributions on a semisimple Lie algebra
cites work: Q3671564
cites work: Q3868000
cites work: Harmonic analysis on real reductive groups
cites work: Analytic Extensions of Differentiable Functions Defined in Closed Sets
OpenAlex ID: W2912608045
description: scientific article; zbMATH DE number 549842