Berezin Quantization, Conformal Welding and the Bott-Virasoro Group.

Following Nag-Sullivan, we study the representation of the group of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space.