Simple Virasoro modules which are locally finite over a positive part

Abstract

Abstract. We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This reduces the problem of classification of simple Virasoro modules which are locally finite over a positive part to classification of 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.

Tensor product weight modules over the Virasoro algebra

“…Later a 4-parameter family of simple weight modules with infinitely dimensional weight spaces were given in [CM] in 2001. Recently in [LLZ] another huge class of simple modules with infinitely dimensional weight spaces were obtained using non-weight modules introduced in [MZ2].…”

Simple Virasoro modules induced from codimension one subalgebras of the positive part

“…Put together, these two classes exhaust all simple weight V-modules with finite dimensional weight spaces, see [Mt], and even those containing a finite dimensional weight space, see [MZ1]. We also refer the reader to the recent monograph [IK] for a detailed survey of the classical part of the representation theory of V. There are a number of other examples of simple V-modules constructed in [Zh,OW,GWZ,LGZ,MZ2] using various tricks. The present note contributes with a new trick leading to a new family of examples depending on five parameters.…”

“…. for some k were studied in [MZ2]. All such modules fit into the general Whittaker setup for V defined in [BM].…”

Whittaker modules for the affine Lie algebraA1(1)

“…Proof of Theorem 4.1 (2)- (4 Recently, Whittaker modules over various Lie algebras are attracting a lot of attention from mathematicians. See [MZ2] and the references there. In Block's classification of all simple modules for the three-dimensional simple Lie algebra sl 2 , they fall into two families: highest (lowest) weight modules, and a family which are simple modules over a Borel subalgebra of sl 2 including Whittaker modules (see [B]).…”

“…Complete and more general results were obtained in [GLZ,MZ2]. The paper [LZ] completely investigated Whittaker modules over the HeisenbergVirasoro algebra.…”