Special Transverse Slices and Their Enveloping Algebras

Abstract

Let G be a simple, simply connected algebraic group over C , g =Lie G, N ( g ) the nilpotent cone of g , and ( E, H, F) an s l 2-triple in g . Let S= E+Ker ad F, the special transverse slice to the adjoint orbit Ω of E, and S 0= S∩ N ( g ). The coordinate ring C [ S 0] 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( g ) be the centre of the enveloping algebra U( g ) and η: Z( g )→ C an algebra homomorphism. Identify g with g * via a Killing isomorphism and let χ denote the linear function on g 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 m χ ⊂ g of dimension (dim Ω)/2 and a 1-dimensional m χ -module C χ . Let H̃ χ denote the algebra opposite to End g ( U( g )⊗ U( m χ) C χ ) and H̃ χ,η = H̃ χ ⊗ Z( g ) C η . It is proved in the paper that the algebra H̃ χ,η has a natural filtration such that gr( H̃ χ,η ), the associated graded algebra, is isomorphic to C [ S 0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of g .