Coadjoint orbits for the central extension of Diff⁺(S ¹) and their representatives

According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits script O sign. In the forward direction one applies the methods of geometric quantization to produce a representation, and in the reverse direction one computes a transform of the character of a representation, to obtain a coadjoint orbit. The method of orbits in the representations of Lie groups suggests the detailed study of coadjoint orbits of a Lie group G in the space script G sign* dual to the Lie algebra script G sign of G. In this paper, two primary goals are achieved: one is to completely classify the smooth coadjoint orbits of Virasoro group for nonzero central charge c; the other is to find representatives for coadjoint orbits. These questions have been considered previously by Segal, Kirillov, and Witten, but their results are not quite complete. To accomplish this, the authors start by describing the coadjoint action of D-the Lie group of all orientation preserving diffeomorphisms on the circle S¹, and its central extension D̄, then the authors will give a complete classification of smooth coadjoint orbits. In fact, they can be parameterized by a subspace of conjugacy classes of PSU(1, 1). Finally, the authors will show how to find representatives of coadjoint orbits by analyzing the vector fields stabilizing the orbits, and describe the amazing connection between the characteristic (trace) of conjugacy classes of PSU(1, 1) and that of vector fields stabilizing orbits.