Abstract: Let M be a compact connected symplectic manifold endowed with a Hamiltonian torus action and associated momentum map $\Phi$. The Atiyah-Guillemin-Sternberg convexity theorem asserts that $\Phi(M)$ is a convex polyhedron.

For a Lagrangian submanifold Q of M we are interested in the image $\Phi(Q)$. In this context Duistermaat extended the AGS-theorem and determined a class of Lagrangians for which $\Phi(Q)=\Phi(M)$ holds.

The objective of this talk is to explain how one can further enlarge Duistermaat's class of Lagrangians and still preserve $\Phi(M)=\Phi(Q)$. One obtains useful applications to classical eigenvalue problems and their Lie theoretic generalizations. In particular we will explain how one can prove Kostant's non-linear convexity theorem with symplectic methods.

We report on joint work with Michael Otto.