Infinitely differentiable functions invariant on the tangent space of a symmetric space
type: Item
instance of: scholarly article
title: Infinitely differentiable functions invariant on the tangent space of a symmetric space
zbMATH Open document ID: 0914.22011
author: Nouri Kamoun
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1999-06-21T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73454
review text: Let \(G\) be a connected reductive Lie group, \(\sigma\) an involution of \(G\) and \(H\) the identity component of the group of its fixed points under \(\sigma\). Let \({\mathfrak g}\) be the Lie algebra of \(G\) and \({\mathfrak g}= {\mathfrak h}+{\mathfrak q}\) its decomposition into \(+1\)(resp., \(-1)\)-eigenspaces for \(\sigma\). Use Harish-Chandra's descent method from the Lie algebra to the tangent space of a symmetric space. The author obtains a necessary and sufficient condition for a function \(f\) defined on \({\mathfrak q}\) to be \(C^\infty\) and \(H\)-invariant, that is: (1) the restriction of \(f\) on any Cartan subspace \({\mathfrak a}\) of \({\mathfrak q}\) remains \(C^\infty\) and is \(W(H,{\mathfrak a})\)-invariant, where \(W(H,{\mathfrak a})\) is the Weyl group of \(H\) in \({\mathfrak a}\); (2) roughly speaking, the restrictions of \(f\) on two different Cartan subspaces of \({\mathfrak q}\) have the same derivatives if these two Cartan subspaces are properly connected. - If \(G\) takes the form \(G'\times G'\), \(G'\) is a reductive Lie group, and the involution \(\sigma:G'\times G'\to G'\times G'\) is defined by \(\sigma (x,y)=(y,x)\), then the above characterization of the \(C^\infty\)-function \(f\) on \({\mathfrak q}\) reduces to a result of \textit{A. Bouaziz} [C. R. Acad. Sci., Paris, Sér. I, 314, 9-12 (1992; Zbl 0784.43007)].
reviewed by: Fuliu Zhu
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 1240926
zbMATH Keywords: infinitely differentiable function
zbMATH Keywords: Cartan subgroup
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Harish-Chandra's descent method
zbMATH Keywords: symmetric space
MaRDI profile type: Publication
OpenAlex ID: W2316671634
cites work: Q3991870
cites work: On the \(C^\infty\) Chevalley's theorem
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DOI: 10.5802/AFST.899
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description: scientific article; zbMATH DE number 1240926