THE SYMMETRIC INVARIANTS OF CENTRALIZERS AND SLODOWY GRADING II

Let g be a finite-dimensional simple Lie algebra of rank ℓ over an algebraically closed field k of characteristic zero, and let (e, h, f) be an sl2-triple of g. Denote by g^e the centralizer of e in g and by S(g^e)^ge the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some ℓ homogenous elements of S(g^e)^ge in (g^e)* has codimension ℓ. If e is good then S(g^e)^ge is a polynomial algebra. In this paper, we prove that the converse of the main result of [CM16] is true. Namely, we prove that e is good if and only if for some homogenous generating sequence q1 ,. .. , qℓ of S(g)^g , the initial homogenous components of their restrictions to e + g^f are algebraically independent over k