Polar Representations and Symmetric Spaces

An orthogonal representation of a compact Lie group K on a euclidean vector space V is called polar if there exists a linear subspace ("section") which intersects each orbit perpendicularly. Examples are the isotropy representations of Riemannian symmetric spaces G/K where the section is any maximal flat subspace. J. Dadok has shown a converse statement: any polar representation is orbit equivalent to the isotropy representation of a Riemannian symmetric space, i.e. there exists an orthogonal Lie triple R : V ⨯ V ⨯ V → V such that the K-orbits on V agree with the orbits of the orthogonal automorphism group of (V,R). In fact, Dadok obtains this result by classifying all polar representations. We give a direct proof for the case where the cohomogeneity is bigger than 2: We construct the Lie triple product R on V from the submanifold geometry of the K-orbits.

(joint work with E.Heintze)