Abstract:
Actions of Lie groups by symplectic transformations are quite often Hamiltonian. By definition, this means that there exists an equivariant momentum map on the manifold with values in the dual Lie algebra of the acting group. In this project, we mainly focus on multiplicity-free manifolds (e.g. toric varieties, flag varieties and more generally spherical varieties), representation spaces of quivers and Nakajima quiver varieties. In the cases under study, the momentum map has remarkable convexity properties. One of our goals is to understand the images of these momentum maps in terms of the geometry of the varieties and vice versa.