Enveloping algebras of Slodowy slices and the Joseph ideal

  • Alexander Premet

    The University of Manchester, United Kingdom


Let be a simple algebraic group over an algebraically closed field \( \k \) of characteristic , and \( \g=\text{Lie}\,G \). Let be an \( \sl_2 \)-triple in \( \g \) with being a long root vector in \( \g \). Let be the -invariant bilinear form on \( \g \) with and let \( \chi\in\g^* \) be such that for all \( x\in\g \). Let be the Slodowy slice at through the adjoint orbit of and let be the enveloping algebra of ; see [\cite{P02}]. In this note we give an explicit presentation of by generators and relations. As a consequence we deduce that contains an ideal of codimension which is unique if \( \g \) is not of type . 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 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 and apply earlier results of Mili{\v c}i{\'c}--Soergel and Backelin to express the composition multiplicities of the Verma modules for 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 -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