Simple Virasoro modules which are locally finite over a positive part

Abstract

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This construction enables us to classify all simple Virasoro modules that are locally finite over a positive part. To obtain those irreducible Virasoro modules, we use simple modules over a family of finite dimensional solvable Lie algebras. For one of these algebras, all simple modules are classified by R. Block and we extend this classification to the next member of the family. As a result, we recover many known but also construct a lot of new simple Virasoro modules. We also propose a revision of the setup for study of Whittaker modules.