Let (M, ω) be a Hamiltonian G-space with proper momentum map J : M → g *. It is well-known that if zero is a regular value of J and G acts freely on the level set J −1 (0), then the reduced space M 0 := J −1 (0)/G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M 0 is a union of symplectic manifolds, i.e., it is a stratified symplectic space. Arms et al., [2], proved that M 0 possesses a natural Poisson bracket. Using their result we study Hamiltonian dynamics on the reduced space. In particular we show that Hamiltonian flows are strata-preserving and give a recipe for a lift of a reduced Hamiltonian flow to the level set J −1 (0). Finally we give a detailed description of the stratification of M 0 and prove the existence of a connected open dense stratum.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The "quantization commutes with reduction" theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M , provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

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. Academic Press

Abstract. I prove the existence of slices for an action of a reductive complex Lie group on a Kähler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the "relative" Schubert calculus of the flag varieties of the two groups. Contents 1. Introduction 1 2. Subgroups and Weyl chambers 3 3. Main results 13 4. Semistability 20 5. Examples 24 Appendix A. Flag varieties 30 Appendix B. Notation 32 References 33