Weakly symmetric spaces and spherical varieties
type: Item
instance of: scholarly article
title: Weakly symmetric spaces and spherical varieties
zbMATH Open document ID: 0916.53024
DOI: 10.1007/BF01236659
published in: Transformation Groups
publication date: 1999-07-26T00:00:00Z
review text: More than 40 years ago, \textit{A. Selberg} [J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)] introduced weakly symmetric spaces \((M,g)\) in the following way. Let \((M,g)\) be a connected Riemannian manifold, \({\mathcal I}(M)\) its full isometry group, \(G\) a subgroup of \({\mathcal I}(M)\) acting transitively on \(M\), and \(u\) an element of \({\mathcal I}(M)\) satisfying (1) \(uGu^{-1}=G\), (2) \(u^2\in G\), (3) for any two points \(x,y\in M\) there exists \(g\in G\) such that \(gx=\mu y\), \(gy=\mu x\). All symmetric spaces are trivially weakly symmetric spaces but only few non-symmetric examples were known. Some years ago, J. Berndt and the reviewer proved that \((M,g)\) is weakly symmetric with respect to \(G={\mathcal I}(M)\) if and only if for each two points \(x,y\) of \(M\) there exists an isometry interchanging them [\textit{J. Berndt} and \textit{L. Vanhecke}, J. Math. Soc. Japan 48, 745-760 (1996; Zbl 0877.53027)]. Since then, the whole subject has been revitalized and many new nontrivial examples have been discovered. The paper under review is also devoted to the study of weakly symmetric homogeneous spaces. The authors start with the following Definition. Let \(G\) be a connected Lie group and \(\widetilde G=(G,u)\) an extension of index 2. Then (1) and (2) hold. Let \(H\) be a compact subgroup of \(G\). Then \(M=G/H\) is said to be weakly symmetric with respect to \(G\) and \(\mu\) if the action of \(G\) extends to an action of \(\widetilde G\) so that (3) holds. The main result in this paper shows that weakly symmetric homogeneous spaces of real reductive algebraic groups are closely related to spherical varieties. More specifically, they prove the following. Let \(H\) be a reductive algebraic subgroup of a connected algebraic group \(G\) and let \(X=G/H\). Then \(X\) is weakly symmetric with respect to \(G\) and some \(\mu\) if and only if \(X\) is a spherical variety. In his paper quoted above, A. Selberg proved that each weakl…
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1286196
zbMATH Keywords: weakly symmetric homogeneous spaces of real reductive algebraic groups
zbMATH Keywords: spherical varieties
author: Dmitri Akhiezer
author: È. B. Vinberg
MaRDI profile type: Publication
Wikidata QID: Q64115208
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full work available at URL: https://doi.org/10.1007/bf01236659
OpenAlex ID: W2167526052
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description: scientific article; zbMATH DE number 1286196