sentence1: since m is a coadjoint orbit in the coadjoint representation of the virasoro group , one has , as always , the natural kirillov-kostant symplectic form on these homogeneous manifolds .
sentence2: in turn , this dirac bracket defines a lower dimensional non-trivial symplectic form , the kirillov-kostant symplectic form for coadjoint orbits of the virasoro group .