The Kazhdan-Lusztig conjecture for finite W-algebras

Details The Kazhdan-Lusztig conjecture for finite W-algebras The Kazhdan-Lusztig conjecture for finite W-algebras

1993-12-03

arxiv.org/abs/hep-th/9312016v1

Lett.Math.Phys.35:333-344,1995

Physics

High Energy Physics

High Energy Physics - Theory

11 pages

Scientific paper

10.1007/BF00750840

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.

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