The symmetric invariants of the centralizers and finite W-algebras

Jean-Yves Charbonnel (1) , Anne Moreau (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. The proof is strongly based on the theory of finite W-algebras. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D7.

Mots clés: symmetric invariant; centralizer; polynomial algebra; finite W-algebra

Domaines: Théorie des représentations [math.RT]

Jean-Yves Charbonnel, Anne Moreau. The symmetric invariants of the centralizers and finite W-algebras. 2013. ⟨hal-00866356⟩