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.