# Enveloping algebras of Slodowy slices and the Joseph ideal

### Alexander Premet

The University of Manchester, United Kingdom

## Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $0$, and $g=LieG$. Let $(e,h,f)$ be an $sl_{2}$-triple in $g$ with $e$ being a long root vector in $g$. Let $(⋅,⋅)$ be the $G$-invariant bilinear form on $g$ with $(e,f)=1$ and let $χ∈g_{∗}$ be such that $χ(x)=(e,x)$ for all $x∈g$. Let $S$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $S$; 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 $g$ is not of type $A$. 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 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 $W$-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.

## Cite this article

Alexander Premet, Enveloping algebras of Slodowy slices and the Joseph ideal. J. Eur. Math. Soc. 9 (2007), no. 3, pp. 487–543

DOI 10.4171/JEMS/86