Gelfand-Kirillov conjecture and Harish-Chandra modules for finite W-algebras

V. Futorny, A. Molev and S. Ovsienko

Abstract

We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker modules of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Harish-Chandra modules for the finite W-algebras.