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

TL;DR

This paper explores the geometric realization of the coadjoint representation of Virasoro-type Lie algebras using Kirillov's method, connecting it to differential operators on tensor densities and illustrating its universality across various infinite-dimensional Lie algebras.

Contribution

It provides a detailed exposition of Kirillov's method for realizing coadjoint representations via linear differential operators, extending its application to Lie superalgebras and other infinite-dimensional Lie algebras.

Findings

Coadjoint representation linked to Sturm-Liouville operators

Kirillov's method is universal for various Lie algebras

Connection to projective differential geometry

Abstract

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.