Enveloping algebras of Slodowy slices and Goldie rank

Let U(g, e) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra g = Lie(G) and let I be a primitive ideal of the enveloping algebra U(g) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that I = AnnU(g) (Qe ⊗U(g, e) V) for some finite dimensional irreducible U(g, e)-module V, where Qe stands for the generalised Gelfand-Graev g-module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient U(g)/I always divides dim V. For g = sln we use a theorem of Joseph on Goldie fields of primitive quotients of U(g) to establish the equality rk(U(g)/I) = dim V. We show that this equality continues to hold for g provided that the Goldie field of g sln is isomorphic to a Weyl skew-field and use this result to disprove Joseph's version of the Gelfand-Kirillov conjecture formulated in the mid-1970s. © 2011 Birkhäuser Boston.