Enveloping algebras of Slodowy slices and the Joseph ideal

By A. Premet. 2005

122 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94a7a8ac11d84567d02748a42d89c9170af82f2d

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Citations in the corpus

On the structure of the category O for W-algebras (2008, background): “ter of Wcan be identiﬁed with the center Zof U, as follows. It is clear that Z⊂Uadm whence we have a homomorphism Z→W. This homomorphism is injective and its image coincides with the center of W, see [Pr2]. An alternative description of Wwas given in [Lo1]. Deﬁne the Slodowy slice S := e + z g(f). I”
Commutative quotients of finite W-algebras (2008, methodology): “In this setting, we have to produce sufficiently many primes p for which the reduction procedure described in [34] leads to irreducible representations of the p-Lie algebra gZ⊗Z Fp with p-characters belonging to the modular counterpart of our initial nilpotent orbit; see Section 4.”
Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups (2011, methodology): “Moreover, in [33] he considers the case where e is a long root element and gives an explicit presentation of Hχ by generators and relations; he further applies this to study the representation theory of Hχ.”
Quantized symplectic actions and W -algebras (2007, background): “Further, if e is minimal, then the existence of a one-dimensional module was proved by Premet in [ Pr2 ].”
Primitive ideals, non-restricted representations and finite W-algebras (2006, unknown): “The action of m on Cχ = C1χ is given by x(1χ) = χ(x)1χ for all x ∈ m; see [18], [11], [6], [20] for more detail.”