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

Abstract

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.