The Kazhdan–Lusztig conjecture for 𝒲 algebras

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, which constructs the 𝒲 algebras from affine Kac–Moody (KM) 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.