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

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

2008-01-07

arxiv.org/abs/0801.0903v3

Mathematics

Rings and Algebras

Scientific paper

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.

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