Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 273–339 | Cite as

The symmetric invariants of centralizers and Slodowy grading

Article

Abstract

Let \(\mathfrak {g}\) be a finite-dimensional simple Lie algebra of rank \(\ell \) over an algebraically closed field \(\Bbbk \) of characteristic zero, and let e be a nilpotent element of \(\mathfrak {g}\). Denote by \(\mathfrak {g}^{e}\) the centralizer of e in \(\mathfrak {g}\) and by \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) the algebra of symmetric invariants of \(\mathfrak {g}^{e}\). We say that e is good if the nullvariety of some \(\ell \) homogenous elements of \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) in \(({\mathfrak g}^{e})^{*}\) has codimension \(\ell \). If e is good then \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) is a polynomial algebra. The main result of this paper stipulates that if for some homogenous generators of \( \mathrm{S}({\mathfrak g}) ^{{\mathfrak g}} \), the initial homogenous components of their restrictions to \(e+\mathfrak {g}^{f}\) are algebraically independent, with (ehf) an \(\mathfrak {sl}_2\)-triple of \(\mathfrak {g}\), then e is good. As applications, we pursue the investigations of Panyushev et al. (J. Algebra 313:343–391, 2007) and we produce (new) examples of nilpotent elements that satisfy the above polynomiality condition, in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type \(\mathbf{D}_{7}\).

Keywords

Symmetric invariant Centralizer Polynomial algebra Slodowy grading 

Mathematics Subject Classification

17B35 17B20 13A50 14L24 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Groupes, Représentations et géométrie, Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586Université Paris Diderot - CNRSParis Cedex 13France
  2. 2.Laboratoire de Mathématiques et Applications de Poitiers (LMA)Futuroscope Chasseneuil CedexFrance

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