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2006.289: Special transverse slices and their enveloping algebras

2006.289: Alexander Premet (2002) Special transverse slices and their enveloping algebras. Advances in Mathematics, 170 (1). pp. 1-55. ISSN 0001-8708

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DOI: 10.1006/aima.2001.2063


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
Uncontrolled Keywords:universal enveloping algebra; deformation quantisation.
Subjects:MSC 2000 > 20 Group theory and generalizations
MIMS number:2006.289
Deposited By:Ms Lucy van Russelt
Deposited On:15 August 2006

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