Symplectic Stratified Spaces and Reduction

Given a Hamiltonian G-space (M,ω,A, μ), let us consider the topological subspace μ−1(0) of M . Since 0 ∈ g∗ is a fixed point of the coadjoint representation, and since μ is G-equivariant, it follows that A restricts to a G-action on μ−1(0). Accordingly, we may consider the quotient topological space M0 := μ−1(0)/G, called the reduced space of (M,ω,A, μ). In the presence of certain additional hypotheses, M0 is naturally a symplectic manifold. However, this will not hold for the general Hamiltonian G-space. Nevertheless, if one requires G to be compact, then M0 will have intriguing topological properties. In particular, there is a partition of M0 into symplectic manifolds fitting together in some desirable ways. This partition realizes M0 as a so-called symplectic stratified space. We will develop the notions necessary to formulate a precise definition of this object, and we will subsequently exhibit M0 as a symplectic stratified space.