Combinatorial invariants of algebraic Hamiltonian actions

To any Hamiltonian action of a reductive algebraic group $G$ on a smooth irreducible symplectic variety $X$ we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. For cotangent bundles our invariants essentially coincide with those arising in the theory of equivarant embeddings. Using our approach we establish some properties of the latter invariants.