Classification of simple $W_n$-modules with finite-dimensional weight spaces

TL;DR

This paper classifies all simple modules of the Lie algebra $W_n$ with finite-dimensional weight spaces, extending classical results and confirming a conjecture by Eswara Rao through a new functor and algebraic identities.

Contribution

It provides a complete classification of simple $W_n$-modules with finite-dimensional weight spaces, including a new functor and algebraic identities that advance understanding of these modules.

Findings

All such modules are either highest weight or quotients of tensor field modules.

Constructed a functor linking cuspidal modules to modules with torus function actions.

Introduced a new identity for quadratic elements in the universal enveloping algebra of $W_1$.

Abstract

We classify all simple $W_n$-modules with finite-dimensional weight spaces. Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao. This generalizes the classical result of Mathieu on simple weight modules for the Virasoro algebra. In our proof of the classification we construct a functor from the category of cuspidal $W_n$-modules to the category of $W_n$-modules with a compatible action of the algebra of functions on a torus. We also present a new identity for certain quadratic elements in the universal enveloping algebra of $W_1$, which provides important information about cuspidal $W_1$-modules.