Proper group actions and symplectic stratified spaces

Let $(M,\omega )$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$\mathrm{.}Itiswell-knownthatif$\alpha$isaregularvalueof$F$and$G$actsfreelyandproperlyonthelevelset$F^{-1}(G\cdot \alpha)$, then the reduced space $M_{\alpha }:=F^{-1}(G\cdot \alpha )\mathrm{/}G$ is a symplectic manifold. We show that if the regularity assumptions are dropped the space $M_{\alpha }$ 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 {\em symplectic stratified space}. This extends results known for the Hamiltonian action of compact groups.