The aim of this talk is to describe the structure symplectic algebraic varieties equipped with an action of a reductive group and with a moment map (Hamiltonian varieties) that contain invariant Lagrangian subvarieties. Most natural examples are cotangent bundles of varieties with a reductive group action, where the zero section is an invariant Lagrangian subvariety. We show that some important invariants of a Hamiltonian symplectic variety with an invariant Lagrangian subvariety coincide with those of the cotangent bundle over this subvariety.