Mathematische Annalen

, Volume 329, Issue 1, pp 49–85 | Cite as

Nilpotent orbits over ground fields of good characteristic

  • George J. McNinch


Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).


Bilinear Form Good Characteristic Reductive Group Nilpotent Element Nilpotent Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bardsley, P., Richardson, R.W.: Étale slices for algebraic transformation groups in characteristic p. Proc. London Math. Soc., 51(3), 295–17 (1985). MR86m:14034Google Scholar
  2. 2.
    Borel, A., Springer, T.A.: Rationality properties of linear algebraic groups. II, Tôhoku Math. J. 20(2), 443–497 (1968) MR39 # 5576Google Scholar
  3. 3.
    Deligne, P., Kazhdan, D., Vignéras, M.-F.: Représentations des algèbres centrales simples p-adiques. Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33–117. MR 86h:11044 (French)Google Scholar
  4. 4.
    Humphreys, J.E.: Conjugacy classes in semisimple algebraic groups. Math. Surveys and Monographs. Am. Math. Soc. 43, 1995Google Scholar
  5. 5.
    Jantzen, J.C.: Nilpotent orbits in representation theory. In: Anker, J.-P. and Orsted, B. (eds.), Lie Theory: Lie Algebras and Representations. Progress in Mathematics, vol. 228, Birkhäuser, Boston, 2004, pp. 1–211Google Scholar
  6. 6.
    Knus, M.A., Merkurjev, A., Rost, M., Tignol, J.P.: The book of involutions. Am. Math. Soc. Colloq. Publ., vol. 44, Am. Math. Soc., 1998Google Scholar
  7. 7.
    Kempf, G.R.: Instability in invariant theory. Ann. of Math. 108(2), 299–316 (1978) MR80c:20057zbMATHGoogle Scholar
  8. 8.
    Laumon, G.: Cohomology of Drinfeld modular varieties. Part I, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, Cambridge, 1996, ISBN= 0-521-47060-9, Geometry, counting of points and local harmonic analysis. MR98c:11045aGoogle Scholar
  9. 9.
    McNinch, G.J.: Sub-principal homomorphisms in positive characteristic. Math. Z. 244, 433–455 (2003)zbMATHGoogle Scholar
  10. 10.
    McNinch, G.J.: Optimal SL(2)-homomorphisms, (2003) math.RT/0309385Google Scholar
  11. 11.
    McNinch, G.J., Sommers, E.: Component groups of unipotent centralizers in good characteristic. J. Alg. 260, 323–337, (2003) arXiv:math.RT/0204275CrossRefzbMATHGoogle Scholar
  12. 12.
    Morris, L.: Rational conjugacy classes of unipotent elements and maximal tori, and some axioms of Shalika. J. London Math. Soc. 38(2), 112–124 (1988) MR89j:22037zbMATHGoogle Scholar
  13. 13.
    Platonov, V., Rapinchuk, A.: Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, 1994, English translationGoogle Scholar
  14. 14.
    Premet, A. Nilpotent orbits in good characteristic and the Kempf-Rousseau theory. J. Alg 260, 338–366 (2003)Google Scholar
  15. 15.
    Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tohoku Math. J. 36(2), 269–291 (1984) MR85j:14017zbMATHGoogle Scholar
  16. 16.
    Ranga~Rao, R.: Orbital integrals in reductive groups. Ann. of Math. 96(2), 505–510 (1972) MR47 # 8771Google Scholar
  17. 17.
    Serre, J.-P.: Lie algebras and Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR36 # 1582Google Scholar
  18. 18.
    Serre, J.-P.: Local fields, Grad. Texts in Math. Springer Verlag, 67, 1979Google Scholar
  19. 19.
    Serre, J.-P.: Galois cohomology, Translated from the French by Patrick Ion and revised by the author, Springer-Verlag, Berlin, 1997, ISBN 3-540-61990-9, MR 98g:12007Google Scholar
  20. 20.
    Springer, T.A.: Linear algebraic groups, 2nd ed., Progr. in Math.,vol. 9, Birkhäuser, Boston, 1998Google Scholar
  21. 21.
    Springer, T.A., Steinberg, R.: Conjugacy classes, Seminar on algebraic groups and related finite groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Springer, Berlin, 1970, pp. 167–266, Lecture Notes in Mathematics, Vol. 131 MR42 # 3091Google Scholar
  22. 22.
    Steinberg, R.: Conjugacy classes in algebraic groups, Springer-Verlag, Berlin, 1974, Notes by Vinay V. Deodhar, Lecture Notes in Mathematics, Vol. 366, MR50 # 4766Google Scholar
  23. 23.
    Tits, J.: Reductive groups over local fields, (Borel, A, Casselman, W) Proc. Sympos. Pure Math., vol. XXXIII Am. Math. Soc., Providence, RI, pp. 29–69Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameUSA

Personalised recommendations