Low cohomogeneity representations and orbit maximal actions

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