Proper group actions and symplectic stratified spaces

TL;DR

This paper extends the theory of symplectic reduction to cases where regularity conditions are dropped, showing that the reduced space forms a symplectic stratified space rather than a smooth manifold.

Contribution

It generalizes the concept of symplectic reduction to include non-regular cases, demonstrating that the reduced space is a symplectic stratified space.

Findings

Reduced spaces are unions of symplectic manifolds.

The symplectic manifolds are organized in a stratified structure.

Extension of known results for compact group actions.

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.