Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture

TL;DR

This paper classifies all momentum proper exact Hamiltonian actions of compact Lie groups, extending the Eliashberg cotangent bundle conjecture and linking symplectic representations with Hamiltonian actions.

Contribution

It provides a complete classification of momentum proper Hamiltonian G-actions, connecting symplectic representations and the Eliashberg cotangent bundle conjecture.

Findings

Classifies all momentum proper exact Hamiltonian G-actions.

Establishes a bijection between symplectic representations and Hamiltonian actions.

Extends the Eliashberg cotangent bundle conjecture to transitive smooth actions.

Abstract

Let $G$ be a compact and connected Lie group. The Hamiltonian $G$-model functor maps the category of symplectic representations of closed subgroups of $G$ to the category of exact Hamiltonian $G$-actions. Based on previous joint work with Y. Karshon, the restriction of this functor to the momentum proper subcategory on either side induces a bijection between the sets of isomorphism classes. This classifies all momentum proper exact Hamiltonian $G$-actions (of arbitrary complexity). As an extreme case, we obtain a version of the Eliashberg cotangent bundle conjecture for transitive smooth actions. As another extreme case, the momentum proper Hamiltonian $G$-actions on contractible manifolds are exactly the symplectic $G$-representations, up to isomorphism.