The Kazhdan-Lusztig conjecture for finiteW-algebras

Abstract

We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-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 finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-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 infiniteW-algebras.