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 maxi

mal 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 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</

I>-orbits.

(joint work with E.Heintze)