The Virasoro Lie algebra is the non-trivial central extension of the Lie
algebra of vector fields on the circle. There is a well-known 1-1
correspondence between the coadjoint orbits in the affine dual of the
Virasoro algebra and conjugacy classes in a certain open subset of the
universal cover of SL(2,R). We extend this to a geometric correspondence,
taking into account the Lie-Poisson structure. As an application, one
obtains a correspondence between Hamiltonian Virasoro spaces and
quasi-Hamiltonian spaces for the universal cover of SL(2,R). (Joint work
with Anton Alekseev.)