Low cohomogeneity representations and orbit maximal actions
type: Item
instance of: scholarly article
title: Low cohomogeneity representations and orbit maximal actions
zbMATH Open document ID: 1029.22018
DOI: 10.1023/A:1021259105021
author: Andreas Kollross
published in: Annals of Global Analysis and Geometry
publication date: 2003-04-27T00:00:00Z
review text: Let \(M\) be a Riemannian manifold and let an isometric action of a compact connected Lie group \(G\) on \(M\) be given. An isometric action of another compact Lie group \(G'\) on \(M\) is called orbit equivalent to the first one if an isometry of \(M\) exists which maps \(G\)-orbits onto \(G'\)-orbits. The action of \(G\) is called orbit maximal if any isometric action of \(G'\) such that every \(G\)-orbit is contained in a \(G'\)-orbit is either orbit equivalent to it or transitive on \(M\). The action is called polar if there exists an embedded smooth submanifold of \(M\) which intersects the orbits orthogonally and meets every orbit. The action is of low cohomogeneity if the dimension of its principal orbit is \(\geq\binom{k+1}2\), where \(k\) is the cohomogeneity of it. Let \(G\) be a compact connected Lie group and \(\rho: G\to\text{SO}(n)\) an irreducible orthogonal representation. The main result of the paper is the equivalence of the following properties: (i) \(\rho\) is polar; (ii) \(\rho\) is of low cohomogeneity and the restriction of \(\rho\) to the unit sphere is orbit maximal; (iii) \(\rho\) is orbit equivalent to an isotropy representation of a symmetric space. The implication (i) \(\Rightarrow\) (iii) was proved in another way by Dadok who also gave a classification of irreducible polar representations [\textit{J. Dadok}, Trans. Am. Math. Soc. 288, 125-137 (1985; Zbl 0565.22010)].
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1901466
zbMATH Keywords: polar representation
zbMATH Keywords: isotropy representation
zbMATH Keywords: orbit equivalent actions
zbMATH Keywords: orbit maximal action
zbMATH Keywords: low cohomogeneity action
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901466