Gelfand-Kirillov Conjecture and Harish-Chandra Modules for Finite W-Algebras

TL;DR

This paper proves the Gelfand-Kirillov conjecture for finite W-algebras related to general linear Lie algebras and classifies their generic irreducible Harish-Chandra modules, advancing understanding of their structure and representations.

Contribution

It establishes the Gelfand-Kirillov conjecture for finite W-algebras and provides a parametrization and classification of their irreducible Harish-Chandra modules.

Findings

Proof of the Gelfand-Kirillov conjecture for finite W-algebras.

Parametrization of irreducible Harish-Chandra modules by Gelfand-Tsetlin characters.

Complete classification of generic irreducible Harish-Chandra modules.

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 model 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.