Special transverse slices and their enveloping algebras
This paper deals with the singularities arising in Lie theory as the intersection of the nilpotent cone of a Lie algebra and the special slice. It is shown that these singularities admit natural deformations. Let \(G\) be a simply connected algebraic group with Lie algebra \(g\) and \(N(g)\) be the nilpotent cone of \(g\). Let \((E,F,G)\) be an \(sl_2\)-triple in \(g\), denote by \(S_0\) the special transverse slice to the adjoint orbit of \(E\) and by \(X\) the linear function on \(g\) corresponding to \(E\). The author constructs a natural algebra \(H\) naturally associated with \(X\), with a natural filtration such that \(\text{grad}(H)\) is isomorphic to the coordinate ring \(C[S_0]\). This filtration is induced by \textit{C. Moeglin}'s filtration of the enveloping algebra \(U(g)\) [C. R. Acad. Sci., Paris, Sér. I 303, 845-848 (1986; Zbl 0628.17007)]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of \(g\). The author gives some applications of the algebra \(H\) for the theory of \(W\)-algebras. In particular, he conjectures that the BRST quantization of the finite \(W\)-algebra associated with any nilpotent element of \(g\) is isomorphic to \(H\).
scientific article; zbMATH DE number 1836810
scientific article