Low Cohomogeneity Representations and Orbit Maximal Actions

Abstract

An orthogonal representation of a compact Lie group is called polar if thereexists a linear subspace which meets all orbits orthogonally.It has been shown by Conlon that one can associate a Coxeter groupto such a representation.From this, an upper bound for the cohomogeneity of an irreduciblepolar representation can be derived.Another property of irreducible polar representations isthat the action restricted to the unit spherehas maximal orbits in the sense that any action having largerorbits is transitive.We give a classification of orbit maximal actions on spheresand use it to show that irreducible polar representations arecharacterized by these two properties.