Distributions sphériques invariantes sur les espaces symétriques semi-simples \(G_ c/G\). (Invariant spherical distributions over the semi-simple symmetric spaces \(G_ c/G\))
type: Item
instance of: scholarly article
title: Distributions sphériques invariantes sur les espaces symétriques semi-simples \(G_ c/G\). (Invariant spherical distributions over the semi-simple symmetric spaces \(G_ c/G\))
zbMATH Open document ID: 0744.43009
author: Shigeru Sano
published in: Journal of Mathematics of Kyoto University
publication date: 1992-06-27T00:00:00Z
review text: The results of the paper under review were announced [in Lect. Notes Math. 1243, 283-309 (1985; Zbl 0627.43007)] and proved [in RIMS Kôkyûroku 598, 117-180 (1986; Zbl 0675.22006)]. Let \(G\) be a connected semi- simple Lie group, \(H\) the subgroup consisting of the fixed points of an involutive automorphism \(\sigma\) of \(G\). With this involutive automorphism one can associate a map \(\varphi: G\to G\); \(\varphi(g):=g\sigma(g)^{-1}\) and set \(X\) its image in \(G\). Thus \(G/H\) can be included in \(G\) as symmetric \(G\)-space. The problem is to study the invariant spherical distributions on \(X\). There is a natural Weyl formula of integration over \(X\) (Proposition 2.1). The restriction of an invariant spherical distribution on the subset \(X'\) of regular elements of \(X\) is always an analytic function. Conversely, given an analytic function over the subset \(X'\), the author finds in Theorem 11.1 a necessary and sufficient condition, with which this analytic function defines an invariant spherical distribution on \(X\).
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 27578
zbMATH Keywords: connected semi-simple Lie group
zbMATH Keywords: involutive automorphism
zbMATH Keywords: symmetric \(G\)- space
zbMATH Keywords: invariant spherical distributions
zbMATH Keywords: analytic function
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519794
OpenAlex ID: W1600536955
DOI: 10.1215/KJM/1250519794
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description: scientific article; zbMATH DE number 27578