Proper group actions and symplectic stratified spaces

Let (M, w) be a Hamiltonian G-space with a momentum map F : M → g*. It is well-known that if α is a regular value of F and G acts freely and properly on the level set F^-1 (G · α), then the reduced space M_α := F^-1(G · α)/G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M_α is a union of symplectic manifolds, and that the symplectic manifolds fit together in a nice way. In other words the reduced space is a symplectic stratified space. This extends results known for the Hamiltonian action of compact groups.