Stratified Symplectic Spaces and Reduction

Let (M, w) 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 '(0), then the reduced space MO = J- '(0)/G is a symplectic manifold. We show that if the regularity assumptions are dropped, the space MO is a union of symplectic manifolds; i.e., it is a stratified symplectic space. Arms et al. 2 proved that MO 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 lifting a reduced Hamiltonian flow to the level set J-'(0). Finally we give a detailed description of the stratification of MO and prove the existence of a connected