Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

We obtain the full classification of coisotropic and polar actions of compact Lie group on irreducible Hermitian symmetric spaces.Comment: 19 pages, no figures final versio

Convexity properties of gradient maps associated to real reductive representations

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson. A similar result is proved for the gradient map associated to the natural $G$ action on P(V). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic resul

Properly Discontinuous Isometric Actions on the Unit Sphere of Infinite Dimensional Hilbert Spaces

We study the properly discontinuous and isometric actions on the unit sphere of infinite dimensional Hilbert spaces and we get some new examples of Hilbert manifold with costant positive sectional curvature. We prove some necessary conditions for a group to act isometrically and properly discontinuously and in the case of finitely generated Abelian groups, the necessary and sufficient conditions are given.Comment: 11 pages, no figuere

Homogeneous bundles and the first eigenvalue of symmetric spaces

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.Comment: Some corrections suggested by the referee. To appear on Annales de l'Institut Fourie

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

Remarks on the abelian convexity theorem

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.Comment: To appear on Proceedings of the American Mathematical Societ