Special transverse slices and their enveloping algebras

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 sI(2)-triple in g. Let S = E + Ker ad F, the special transverse slice to the adjoint orbit Omega of E, and S-0 = S boolean AND 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 eta:Z(g)-->C an algebra homomorphism. Identify g with g* via a Killing isomorphism and let X 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 chi a nilpotent subalgebra m(x) subset of g of dimension (dim Omega)/2 and a 1-dimensional m(chi)-module C-chi. Let (H) over tilde (chi) denote the algebra opposite to End(g)(U(g) circle times(U(mz)) C-chi) and (H) over tilde (chi,eta) = (H) over tilde (chi) circle times(Z(g)) Ceta. It is proved in the paper that the algebra (H) over tilde (chi,eta) has a natural filtration such that gr((H) over tilde (chi,eta)), 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. (C) 2002 Elsevier Science (USA).