  We prove several results on symplectic varieties with a Hamiltonian action of
a reductive group having invariant Lagrangian subvarieties. Our main result
states that the images of the moment maps of a Hamiltonian variety and of the
cotangent bundle over an invariant Lagrangian subvariety coincide. This implies
that the complexity and rank of the Lagrangian subvariety are equal to the half
of the corank and to the defect of the Hamiltonian variety, respectively. This
result generalizes a theorem of Panyushev on the complexity and rank of a
conormal bundle. A simple elementary proof of this theorem is also given in the
paper. A generalization of the above results to some special class of invariant
coisotropic subvarieties is obtained.
