Enveloping algebras of Slodowy slices and the Joseph ideal

  • Alexander Premet

    The University of Manchester, United Kingdom


Let GG be a simple algebraic group over an algebraically closed field \k\k of characteristic 00, and \g=LieG\g=\text{Lie}\,G. Let (e,h,f)(e,h,f) be an \sl2\sl_2-triple in \g\g with ee being a long root vector in \g\g. Let (,)(\,\cdot\,,\,\cdot\,) be the GG-invariant bilinear form on \g\g with (e,f)=1(e,f)=1 and let χ\g\chi\in\g^* be such that χ(x)=(e,x)\chi(x)=(e,x) for all x\gx\in\g. Let S{\mathcal S} be the Slodowy slice at ee through the adjoint orbit of ee and let HH be the enveloping algebra of S{\mathcal S}; see [\cite{P02}]. In this note we give an explicit presentation of HH by generators and relations. As a consequence we deduce that HH contains an ideal of codimension 11 which is unique if \g\g is not of type A\mathrm A. Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(\g)U(\g). Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of HH and the spectrum of all primitive ideals of infinite codimension in U(\g)U(\g) which respects Goldie rank and Gelfand--Kirillov dimension. We study highest weight modules for the algebra HH and apply earlier results of Mili{\v c}i{\'c}--Soergel and Backelin to express the composition multiplicities of the Verma modules for HH 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{\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)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