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

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.