Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.Comment: 7 pages; February 21st, 2006: Statement of main result corrected; other minor change

Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous structure together with the second fundamental form encodes all the information about M, and deduce from this the rigidity result that M is completely determined by the second fundamental form and its covariant derivative, thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed by Heintze and Liu to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of the homogeneous structure. Here an important tool is the introduction of affine root systems of isoparametric submanifolds

Polar symplectic representations

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case.Comment: 19 page