COADJOINT ORBITS FOR THE CENTRAL EXTENSION OF Diff^＋（S^1） AND THEIR REPRESENTATIVES

According to Kirillov′s idea, the irreducible unitary representations of a Liegroup G roughly correspond to the coadjoint orbits O. In the forward direction one ap-plies the methods of geometric quantization to produce a representation, and in the reversedirection one computes a transform of the character of a representation, to obtain a coad-joint orbit. The method of orbits in the representations of Lie groups suggests the detailedstudy of coadjoint orbits of a Lie group G in the space g* dual to the Lie algebra g of G.In this paper, two primary goals are achieved: one is to completely classify the smoothcoadjoint orbits of Virasoro group for nonzero central charge c; the other is to find repre-sentatives for coadjoint orbits. These questions have been considered previously by Segal,Kirillov, and Witten, but their results are not quite complete. To accomplish this, theauthors start by describing the coadjoint action of D-the Lie group of all orientation pre-serving diffeomorphisms on the circle S^1, and its central extension D~, then the authors willgive a complete classification of smooth coadjoint orbits. In fact, they can be parameterizedby a subspace Of conjugacy classes of PSU~(1,1). Finally, the authors will show how to findrepresentatives of coadjoint orbits by analyzing the vector fields stabilizing the orbits, anddescribe the amazing connection between the characteristic (trace) of conjugacy classes of PSU~(1, 1) and that of vector fields stabilizing orbits.