The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

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