Convexity Properties of the Moment Mapping Re-examined

Consider a Hamiltonian action of a compact Lie group on a compact symplectic manifold. A theorem of Kirwan's says that the image of the momentum mapping intersects the positive Weyl chamber in a convex polytope. I present a new proof of Kirwan's theorem, which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold. It also applies to some interesting classes of noncompact or singular Hamiltonian spaces, such as cotangent bundles and complex affine varieties.