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Mathematische Zeitschrift

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

The symmetric invariants of centralizers and Slodowy grading

  • Jean-Yves Charbonnel
  • Anne Moreau
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 

Notes

Acknowledgments

We thank Alexander Premet for his important comments on the previous version of this paper. We also thank the referee for careful reading and thoughtful suggestions. This work was partially supported by the ANR-Project 10-BLAN-0110.

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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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