Enveloping algebras of Slodowy slices and Goldie rank

Abstract

Let $U\left( {\mathfrak{g},e} \right)$ be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra $\mathfrak{g} = {\text{Lie}}(G)$ and let I be a primitive ideal of the enveloping algebra $U\left( \mathfrak{g} \right)$ whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that $I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right)$ for some finite dimensional irreducible $U\left( {\mathfrak{g},e} \right)$-module V, where Q _e stands for the generalised Gelfand–Graev $\mathfrak{g}$-module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient ${{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}$ always divides dim V. For $\mathfrak{g} = \mathfrak{s}{\mathfrak{l}_n}$, we use a theorem of Joseph on Goldie fields of primitive quotients of $U\left( \mathfrak{g} \right)$ to establish the equality ${\text{rk}}\left( {{{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}} \right) = \dim V$. We show that this equality continues to hold for $\mathfrak{g} \ncong \mathfrak{s}{\mathfrak{l}_n}$ provided that the Goldie field of ${{{U\left( \mathfrak{g} \right)}} \left/ {I} \right.}$ 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.