Enveloping algebras of Slodowy slices and the Joseph ideal

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{k}$ of characteristic $0$, and $\mathfrak{g}=\text{Lie}G$. Let $(e,h,f)$ be an ${\mathfrak{sl}}_2$-triple in $\mathfrak{g}$ with $e$ being a long root vector in $\mathfrak{g}$. Let $(\cdot ,\cdot )$ be the $G$-invariant bilinear form on $\mathfrak{g}$ with $(e,f)=1$ and let $\chi \in {\mathfrak{g}}^{\ast }$ be such that $\chi (x)=(e,x)$ for all $x\in \mathfrak{g}$. 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 [31]. 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 $\mathfrak{g}$ is not of type $\mathrm{A}$. Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of $U(\mathfrak{g})$. 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 $U(\mathfrak{g})$ which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra $H$ and apply earlier results of Miličić–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 $U(\mathfrak{g})$. A sequel to this paper will treat modular aspects of this theory.