Proper group actions and symplectic stratified spaces

Abstract

Let $(M,\omega)$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$. It is well-known that if $\alpha$ is a regular value of $F$ and $G$ acts freely and properly on the level set $F^{-1}(G\cdot \alpha)$, then the reduced space $M_\alpha :=F^{-1}(G\cdot \alpha)/G$ 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.