Enveloping algebras of Slodowy slices and the Joseph ideal

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field \( \k \) of characteristic $0$, and \( \g=\text{Lie}\,G \). Let $(e,h,f)$ be an \( \sl_2 \)-triple in \( \g \) with $e$ being a long root vector in \( \g \). Let $(\cdot ,\cdot )$ be the $G$-invariant bilinear form on \( \g \) with $(e,f)=1$ and let \( \chi\in\g^* \) be such that $\chi (x)=(e,x)$ for all \( x\in\g \). Let $\mathcal{S}$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $\mathcal{S}$; see [\cite{P02}]. In this note we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains an ideal of codimension $1$ which is unique if \( \g \) is not of type $\mathrm{A}$. Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of \( U(\g) \). Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of $H$ and the spectrum of all primitive ideals of infinite codimension in \( U(\g) \) which respects Goldie rank and Gelfand--Kirillov dimension. We study highest weight modules for the algebra $H$ and apply earlier results of Mili{\v c}i{\'c}--Soergel and Backelin to express the composition multiplicities of the Verma modules for $H$ in terms of some inverse parabolic Kazhdan--Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos--van Driel conjecture on composition multiplicities of Verma modules for finite $\mathcal{W}$-algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of \( U(\g) \). A sequel to this paper will treat modular aspects of this theory.