The Kazhdan-Lusztig conjecture for W-algebras

TL;DR

This paper proves a character formula for irreducible modules of W-algebras using quantum Hamiltonian reduction and Kazhdan-Lusztig theory, linking W algebra representations to affine Kac-Moody algebra structures.

Contribution

It establishes the Kazhdan-Lusztig conjecture for W-algebras, connecting their representation theory to that of affine Kac-Moody algebras via a functorial approach.

Findings

Derived character formulas for W-algebra modules.

Linked W algebra characters to Kazhdan-Lusztig polynomials.

Provided a geometric interpretation using Weyl group cosets.

Abstract

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of W algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, that constructs the W algebras from affine Kac-Moody algebras and in a similar fashion W modules from KM modules. Assuming certain properties of this functor, the W characters are subsequently derived from the Kazhdan-Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan-Lusztig polynomials give the multiplicities in the characters.