On the classification of polar representations
Let \(V\) be a Euclidean vector space and \(K\subset O(V)\) be a compact subgroup. The representation of \(K\) on \(V\) is said to be polar, if there exists a linear subspace \(\Sigma\) which meets every orbit and meets it perpendicularly at every point of intersection. Examples are the so called \(s\)-representations, i.e. the isotropy representations of symmetric spaces. In [\textit{J. Dadok}, Trans. Am. Math. Soc. 288, 125-137 (1985; Zbl 0565.22010)] all irreducible polar representations are classified almost explicitly. In this paper, the authors show that for an irreducible symmetric space of rank at least 2 with connected \(K\), if a connected closed subgroup \(K'\) of \(K\) has the same orbits as \(K\) under the isotropy representation, then \(K= K'\) unless \((G/K,K')\) is among a certain explicit list of six cases. The main idea of the proof consists in the observation that the principal \(K\)-orbits are isoparametric submanifolds which are foliated by curvature spheres on which \(K\) and also \(K'\) act transitively. This is a severe restriction on \(K'\) and implies in most cases that \(K=K'\).
scientific article; zbMATH DE number 1384883