# Special transverse slices and their enveloping algebras

Premet, Alexander (2002) Special transverse slices and their enveloping algebras. Advances in Mathematics, 170 (1). pp. 1-55. ISSN 0001-8708 PDF GGG.pdf Download (467kB)

## Abstract

Let G be a simple, simply connected algebraic group over Image, Image=Lie G, Image (Image) the nilpotent cone of Image, and (E,H,F) an ImageImage2-triple in Image. Let S=E+Ker ad F, the special transverse slice to the adjoint orbit Ω of E, and S0=S∩Image (Image). The coordinate ring Image[S0] is naturally graded (See Slodowy, "Simple Singularities and Simple Algebraic Groups," Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin/Heidelberg/New York, 1980). Let Z(Image) be the centre of the enveloping algebra U(Image) and η:Z(Image)→Image an algebra homomorphism. Identify Image with Image* via a Killing isomorphism and let χ denote the linear function on Image corresponding to E. Following Kawanaka (Generalized Gelfand–Graev representations and Ennola duality, in "Algebraic Groups and Related Topics" Advanced Studies in Pure Mathematics, Vol. 6, pp. 175–206, North-Holland, Amsterdam/New York/Oxford, 1985), Moeglin (C.R. Acad. Sci. Paris, Ser. I 303 No. 17 (1986), 845–848), and Premet (Invent. Math.121 (1995), 79–117), we attach to χ a nilpotent subalgebra Imageχsubset ofImage of dimension (dim Ω)/2 and a 1-dimensional Imageχ-module Imageχ. Let ImageH" height="14" width="14">χ denote the algebra opposite to EndImage(U(Image)multiply sign in circleU(Imageχ)Imageχ) and ImageH" height="14" width="14">χ,η=ImageH" height="14" width="14">χmultiply sign in circleZ(Image)Imageη. It is proved in the paper that the algebra ImageH" height="14" width="14">χ,η has a natural filtration such that gr(ImageH" height="14" width="14">χ,η), the associated graded algebra, is isomorphic to Image[S0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of Image.

Item Type: Article universal enveloping algebra; deformation quantisation. MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations Ms Lucy van Russelt 15 Aug 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/495

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