Nagoya Mathematical Journal

The centralizer of a nilpotent section

George J. McNinch

Full-text: Open access

Abstract

Let $F$ be an algebraically closed field and let $G$ be a semisimple $F$-algebraic group for which the characteristic of $F$ is very good. If $X \in \operatorname{Lie}(G) = \operatorname{Lie}(G)(F)$ is a nilpotent element in the Lie algebra of $G$, and if $C$ is the centralizer in $G$ of $X$, we show that (i) the root datum of a Levi factor of $C$, and (ii) the component group $C/C^{o}$ both depend only on the Bala-Carter label of $X$; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when $G$ is (simple and) of adjoint type.

The proofs are achieved by studying the centralizer $\mathcal{C}$ of a nilpotent section $X$ in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring $\mathcal{A}$. When the centralizer of $X$ is equidimensional on $\operatorname{Spec}(\mathcal{A})$, a crucial result is that locally in the étale topology there is a smooth $\mathcal{A}$-subgroup scheme $L$ of $\mathcal{C}$ such that $L_{t}$ is a Levi factor of $\mathcal{C}_{t}$ for each $t \in \operatorname{Spec}(\mathcal{A})$.

Article information

Source
Nagoya Math. J., Volume 190 (2008), 129-181.

Dates
First available in Project Euclid: 23 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1214229081

Mathematical Reviews number (MathSciNet)
MR2423832

Zentralblatt MATH identifier
1162.20030

Subjects
Primary: 20G15: Linear algebraic groups over arbitrary fields

Citation

McNinch, George J. The centralizer of a nilpotent section. Nagoya Math. J. 190 (2008), 129--181. https://projecteuclid.org/euclid.nmj/1214229081


Export citation

References

  • A. V. Alekseevskiĭ, Component groups of centralizers of unipotent elements in semisimple algebraic groups, Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze, 62 (1979). Collection of articles on algebra, 2.
  • Armand Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math., vol. 126, Springer Verlag, 1991.
  • Roger W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, John Wiley & Sons Ltd., Chichester, 1993. Reprint of the 1985 original.
  • A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math., 28 (1966), 255.
  • A. Grothendieck, Revêtements étales et groupe fondamental (SGA $1$), Documents Mathématiques 3, Société Mathématique de France, Paris, 2003. Séminaire de Géometrie Algébrique du Bois Marie. [Updated and annotated reprint of the 1971 original publication in Lecture Notes in Math., 224, Springer, Berlin.]
  • A. Grothendieck and M. Demazure, Schémas en Groupes (SGA 3). I, II, III, Lectures Notes in Math., vol. 151, 152, 153, Springer Verlag, Heidelberg, 1965. Séminaire de Géometrie Algébrique du Bois Marie.
  • James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Math. Surveys and Monographs, vol. 43, Amer. Math. Soc., 1995.
  • Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie Theory: Lie Algebras and Representations (J.-P. Anker and B. Orsted, eds.), Progress in Mathematics, vol. 228, Birkhäuser, Boston, 2004, pp. 1--211.
  • Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003.
  • George R. Kempf, Instability in invariant theory, Ann. of Math. (2), 108 (1978), no. 2, 299--316.
  • Max-Albert Knus, Alexander Merkurjev, Markus Rost and Jean-Pierre Tignol, The book of involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., 1998.
  • Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, 2002. Translated from the French by Reinie Erné.
  • George J. McNinch and Eric Sommers, Component groups of unipotent centralizers in good characteristic, J. Algebra, 260 (2003), no. 1, 323--337. Special issue celebrating the 80th birthday of Robert Steinberg. math.RT/0204275.
  • George J. McNinch and Donna M. Testerman, Completely reducible $\operatornameSL(2)$-homomorphisms, Trans. Amer. Math. Soc., 359 (2007), no. 9, 4489--4510.
  • George J. McNinch, Nilpotent orbits over ground fields of good characteristic, Math. Annalen, 329 (2004), 49--85. arXiv:math.RT/0209151.
  • --------, Optimal $\operatornameSL(2)$-homomorphisms, Comment. Math. Helv., 80 (2005), 391--426. arXiv:math.RT/0309385.
  • --------, On the centralizer of the sum of commuting nilpotent elements, J. Pure and Applied Algebra, 206 (2006), 123--140. arXiv:math.RT/0412283.
  • James Milne, Étale Cohomology, Princeton University Press, 1980.
  • Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups $E\sb7$ and $E\sb8$, Tokyo J. Math., 3 (1980), no. 2, 391--461.
  • Alexander Premet, Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra, 260 (2003), no. 1, 338--366. Special issue celebrating the 80th birthday of Robert Steinberg.
  • R. W. Richardson, On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc., 25 (1982), no. 1, 1--28.
  • Jean-Pierre Serre, Local Fields, Springer Verlag, 1979.
  • --------, Complète Réductibilité, Astérisque, 299 (2005), Exposés 924--937, pp. 195--217. Séminaire Bourbaki 2003/2004.
  • Eric Sommers, A generalization of the Bala-Carter theorem for nilpotent orbits, Internat. Math. Res. Notices (1998), no. 11, 539--562.
  • Tonny A. Springer, Linear algebraic groups, 2nd ed., Progr. in Math., vol. 9, Birkhäuser, Boston, 1998.
  • Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968.