Non-linear Grassmannians as coadjoint orbits

For a given manifold M we consider the non-linear Grassmann manifold Gr(n)(M) of n-dimensional submanifolds in M. A closed (n+2)-form on M gives rise to a closed 2-form on Gr(n)(M). If the original form was integral, the 2-form will be the curvature of a principal S-1-bundle over Gr(n)(M). Using this S-1-bundle one obtains central extensions for certain groups of diffeomorphisms of M. We can realize Gr(m-2)(M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians SGr(2k)(M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms.