The Kazhdan-Lusztig conjecture for finite W-algebras

TL;DR

This paper investigates the representation theory of finite W-algebras, proposing a Kazhdan-Lusztig type conjecture, analyzing their modules, and introducing dual Kazhdan-Lusztig polynomials to understand their structure.

Contribution

It introduces a conjecture for the structure of finite W-algebras, including a new approach using dual Kazhdan-Lusztig polynomials to analyze modules.

Findings

Identification of the structure of Verma modules for W-algebras

Introduction of dual Kazhdan-Lusztig polynomials for W-algebras

Examples supporting the conjectures and potential applications

Abstract

We study the representation theory of finite W-algebras. After introducing parabolic subalgebras to describe the structure of W-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finite W-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors, and find that for W-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infinite W-algebras.