Coadjoint orbits of the Virasoro group

Abstract

The coadjoint orbits of the Virasoro group, which have been investigated by Lazutkin and Pankratova and by Segal, should according to the Kirillov-Kostant theory be related to the unitary representations of the Virasoro group. In this paper, the classification of orbits is reconsidered, with an explicit description of the possible centralizers of coadjoint orbits. The possible orbits are diff(S 1) itself, diff(S 1)/S 1, and diff(S 1)/SL (n)(2,R), withSL (n)(2,R) a certain discrete series of embeddings ofSL(2,R) in diff(S 1), and diffS 1/T, whereT may be any of certain rather special one parameter subgroups of diffS 1. An attempt is made to clarify the relation between orbits and representations. It appears that quantization of diffS 1/S 1 is related to unitary representations with nondegenerate Kac determinant (unitary Verma modules), while quantization of diffS 1/SL (n)(2,R) is seemingly related to unitary representations with null vectors in leveln. A better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of Virasoro representation theory. In the process of investigating Virasoro coadjoint orbits, we observe the existence of left invariant symplectic structures on the Virasoro group manifold. As is described in an appendix, these give rise to Lie bialgebra structures with the Virasoro algebra as the underlying Lie algebra.