# Enveloping algebras of Slodowy slices and the Joseph ideal

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 3, page 487-543
- ISSN: 1435-9855

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topPremet, 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 -

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