Low cohomogeneity representations and orbit maximal actions
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)].
scientific article; zbMATH DE number 1901466