Enveloping algebras of Slodowy slices and the Joseph ideal

Premet, Alexander. "Enveloping algebras of Slodowy slices and the Joseph ideal." Journal of the European Mathematical Society 009.3 (2007): 487-543. <http://eudml.org/doc/277580>.

@article{Premet2007,
abstract = {Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb \{k\}$ of characteristic 0, and $\mathfrak \{g\}=\operatorname\{Lie\}G$. Let $(e,h,f)$ be an $\mathfrak \{s\}\mathfrak \{l\}_2$-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 $\chi \in \mathfrak \{g\}^*$ 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 article 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 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.},
author = {Premet, Alexander},
journal = {Journal of the European Mathematical Society},
language = {eng},
number = {3},
pages = {487-543},
publisher = {European Mathematical Society Publishing House},
title = {Enveloping algebras of Slodowy slices and the Joseph ideal},
url = {http://eudml.org/doc/277580},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Premet, Alexander
TI - Enveloping algebras of Slodowy slices and the Joseph ideal
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 3
SP - 487
EP - 543
AB - Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb {k}$ of characteristic 0, and $\mathfrak {g}=\operatorname{Lie}G$. Let $(e,h,f)$ be an $\mathfrak {s}\mathfrak {l}_2$-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 $\chi \in \mathfrak {g}^*$ 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 article 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 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.
LA - eng
UR - http://eudml.org/doc/277580
ER -