Enveloping algebras of Slodowy slices and Goldie rank

TL;DR

This paper investigates the relationship between Goldie ranks of primitive quotients of universal enveloping algebras and finite-dimensional modules over finite W-algebras, proving divisibility and equality results, and disproving a longstanding conjecture.

Contribution

It establishes a divisibility relation between Goldie ranks and module dimensions, proves equality in type A, and disproves Joseph's conjecture on Goldie field structures.

Findings

Goldie rank divides the dimension of the finite W-algebra module.

In type A, Goldie rank equals the module dimension.

Disproves Joseph's conjecture on Goldie fields of primitive quotients.

Abstract

It is known that any primitive ideal I of U(g) whose associated variety contains a nilpotent element e in its open G-orbit admits a finite generalised Gelfand-Graev model which is a finite dimensional irreducible module over the finite W-algebra U(g,e). We prove that if V is such a model for I, then the Goldie rank of the primitive quotient U(g)/I always divides the dimension of V. For g=sl(n), we use a result of Joseph to show that the Goldie rank of U(g)/I equals the dimension of V and we show that the equality conntinues to hold outside type A provided that the Goldie field of U(g)/I is isomorphic to a Weyl skew-field. As an application of this result, we disprove Joseph's conjecture on the structure of the Goldie fields of primitive quotients of U(g) formulated in the mid-70s.