Stratified symplectic spaces and reduction

Abstract

Let $(M, \omega)$ be a Hamiltonian $G$-space with proper momentum map $J\colon M \to \frak{g}^\ast$. It is well-known that if zero is a regular value of $J$ and $G$ acts freely on the level set $J^{-1}(0)$, then the reduced space $M_0:= J^{-1}(0)/G$ is a symplectic manifold. We show that if the regularity assumptions are dropped, the space $M_0$ is a union of symplectic manifolds; i.e., it is a stratified symplectic space. Arms et al. [2] proved that $M_0$ 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 $J^{-1}(0)$. Finally we give a detailed description of the stratification of $M_0$ and prove the existence of a connected open dense stratum.

Reyer Sjamaar

Eugene Lerman