  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.
