Large Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities

We discuss the geometrical nature of the coadjoint representation of the Virasoro algebra and some of its generalizations. The isomorphism of the coadjoint representation of the Virasoro group to the $Diff(S^1)$-action on the space of Sturm-Liouville operators was discovered by A.A. Kirillov and G. Segal. This deep and fruitful result relates this topic to the classical problems of projective differential geometry (linear differential operators, projective structures on $S^1$ etc.) The purpose of this talk is to give a detailed explanation of the A.A. Kirillov method [14] for the geometric realization of the coadjoint representation in terms of linear differential operators. Kirillov\'s method is based on Lie superalgebras generalizing the Virasoro algebra. One obtains the Sturm-Liouville operators directly from the coadjoint representation of these Lie superalgebras. We will show that this method is universal. We will consider a few examples of infinite-dimensional Lie algebras and show that the Kirillov method can be applied to them. This talk is purely expository: all the results are known.