Algebras and Representation Theory

, Volume 20, Issue 6, pp 1341–1363 | Cite as

The Symmetric Invariants of Centralizers and Slodowy Grading II

  • Jean-Yves Charbonnel
  • Anne MoreauEmail author


Let 𝔤 be a finite-dimensional simple Lie algebra of rank over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤 through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩2-triple of 𝔤. Denote by 𝔤 e the centralizer of e in 𝔤 and by \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) the algebra of symmetric invariants of 𝔤 e . We say that e is good if the nullvariety of some homogeneous elements of \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) in (𝔤 e ) has codimension . In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q 1, … , q l of S(𝔤) 𝔤 , the initial homogeneous components of their restrictions to e + 𝔤 f are algebraically independent over 𝕜.


Symmetric invariant Centralizer Polynomial algebra Slodowy grading 

Mathematics Subject Classification (2010)

17B35 17B20 13A50 14L24 


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The second author is partially supported by the ANR Project GeoLie Grant number ANR-15-CE40-0012. We thank the referee for valuable comments and suggestions.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu - Paris Rive GaucheUniversité Paris Diderot - CNRSParisFrance
  2. 2.Laboratoire de Mathématiques et ApplicationsFuturoscope Chasseneuil CedexFrance
  3. 3.Laboratoire Paul PainlevéVilleneuve d’Ascq CedexFrance

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