Proper group actions and symplectic stratified spaces

L. Bates and E. Lerman

Vol. 181 (1997), No. 2, 201–229

DOI: 10.2140/pjm.1997.181.201

Abstract

Let (M,ω) be a Hamiltonian G-space with a momentum map F : M → g^∗. It is well-known that if α is a regular value of F and G acts freely and properly on the level set F⁻¹(G ⋅ α), then the reduced space M_α := F⁻¹(G ⋅ α)∕G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M_α 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 symplectic stratified space. This extends results known for the Hamiltonian action of compact groups.

Milestones

Received: 25 May 1995

Revised: 17 June 1996

Published: 1 December 1997