Poisson Geometry of Discrete Series Orbits, and Momentum Convexity for Noncompact Group Actions

The main result of this paper is a convexity theorem for momentum mappings J :M → g∗ of certain hamiltonian actions of noncompact semisimple Lie groups. The image of J is required to fall within a certain open subset D of g∗ which corresponds roughly via the orbit method to the discrete series of representations of the group G. In addition, J is required to be proper as a map from M to D. A related but quite different convexity theorem for noncompact groups may be found in [16]. Our result is a first attempt toward placing momentum convexity theorems in a Poisson-geometric setting. A momentum mapping is a Poisson mapping to the dual of a Lie algebra, but it takes more than the Poisson structure on the target manifold even to formulate a convexity theorem. As we explained last year’s Conference Moshé Flato Proceedings [20], it seems that a proper symplectic groupoid is the right extra structure to put on the