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.