Invariant vector fields on a simple Lie algebra under the adjoint action (Q793828)

Let \({\mathfrak g}\) be a complex simple Lie algebra, G the adjoint group of \({\mathfrak g}\) and \({\mathcal S}_ Z\) a transversal slice at a given regular nilpotent element Z of the G-orbit of Z, which can be identified with the quotient space \({\mathfrak h}/W\) of a Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) under the action of the Weyl group W [cf. \textit{B. Kostant,} Am. J. Math. 85, 327-404 (1963; Zbl 0124.268)]. Further the algebra \(I({\mathfrak g})\) of G-invariant polynomials on \({\mathfrak g}\) is isomorphic to the coordinate ring C[\({\mathfrak h}/W]\) of \({\mathfrak h}/W\cong {\mathcal S}_ Z\). In this note, the author generalizes this isomorphism theorem on polynomial functions to vector fields. Namely, he defines the vector fields on \({\mathfrak g}\) such that 1) they are invariant under the adjoint action and 2) the restrictions of them to \({\mathfrak h}/W\cong {\mathcal S}_ Z\) form a system of generators for the Lie algebra of vector fields on \({\mathfrak h}/W\) logarithmic along the discriminant set of \({\mathfrak h}/W\).