Representations of the Virasoro algebra by the orbit method (Q912201)

The aim of this paper is to give a geometric background for the highest weight representations of the Virasoro algebra. Let \(Diff_+ S^ 1\) be the group of diffeomorphisms of the unit circle \(S^ 1\) preserving an orientation, and \(M=Diff_+ S^ 1/Rot S^ 1\) be the corresponding flag manifold, where Rot \(S^ 1\) is the one-dimensional group of rigid rotations of \(S^ 1\). It is known that M can be identified with the set of univalent holomorphic functions on the unit disc and that M admits a two-parameter family of pseudo-Kählerian metrics \(w_{h,c}\) (h,c\(\in {\mathbb{R}})\). Over each pseudo-Kählerian manifold \((M,w_{h,c})\) one can construct an analytic line bundle \(E_{h,c}\). Since the line bundle \(E_{h,c}\) is analytically trivial, the total space can be identified with \(M\times {\mathbb{C}}.\) The authors realize the representation with highest weight (h,c) of the Virasoro algebra in a space of holomorphic sections of \(E_{h,c}\). An explicit formula of the action is also given. Finally the authors describe an embedding of M into an infinite dimensional analogue of the bounded symmetric domain of type III.