Stratification of the transverse momentum map

TL;DR

This paper introduces a natural stratification of the transverse momentum map for Hamiltonian actions of proper symplectic groupoids, refining existing orbit type stratifications and compatible with Poisson geometry.

Contribution

It constructs a new Poisson stratification of the orbit space, compatible with the Poisson structure, and develops a normal form theorem for Hamiltonian actions of proper symplectic groupoids.

Findings

The transverse momentum map admits a natural constant rank stratification.

The refined stratification is compatible with Poisson geometry.

Each stratum is a regular Poisson manifold with a proper symplectic groupoid.

Abstract

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle-Guillemin-Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.