Algebras and Representation Theory

, Volume 14, Issue 1, pp 161–190 | Cite as

On Nilpotent Orbits of SL n and Sp2n over a Local Non-Archimedean Field

Article

Abstract

We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the classical groups SL n and Sp2n over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.

Keywords

Nilpotent orbits DeBacker parametrization Algebraic groups Bruhat-Tits building 

Mathematics Subject Classification 2000

20G25 (17B45) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Reprint of the 1985 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, Chichester (1993)Google Scholar
  2. 2.
    Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold, New York (1993)MATHGoogle Scholar
  3. 3.
    DeBacker, S.: Parametrizing nilpotent orbits via Bruhat-Tits theory. Ann. Math. (2) 156(1), 295–332 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    DeBacker, S.: Homogeneity for reductive p-adic groups: an introduction. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, pp. 523–550. Clay Math. Proc., 4. American Mathematical Society, Providence (2005)Google Scholar
  5. 5.
    Doković, D.Ž.: Note on rational points in nilpotent orbits of semisimple groups. Indag. Math. (N.S.) 9(1), 31–34 (1998)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jantzen, J.C.: Nilpotent orbits in representation theory. In: Lie Theory, pp. 1–211. Progr. Math., 228. Birkhäuser Boston, Boston (2004)Google Scholar
  7. 7.
    Kottwitz, R.E.: Rational conjugacy classes in reductive groups. Duke Math. J. 49(4), 785–806 (1982)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lam, T.Y.: Introduction to quadratic forms over fields. In: Graduate Studies in Mathematics, AMS, vol. 67 (2005)Google Scholar
  9. 9.
    McNinch, G.J.: Nilpotent orbits over ground fields of good characteristic. Math. Ann. 329(1), 49–85 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    McNinch, G.J.: Optimal SL(2)-homomorphisms. Comment. Math. Helv. 80(2), 391–426 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Moy, A., Prasad, G.: Unrefined minimal K-types for p-adic groups. Invent. Math. 116(1–3), 393–408 (1994)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nevins, M.: Admissible nilpotent orbits of real and p-adic split exceptional groups. Representat. Theory 6, 160–189 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Noël, A.G.: Nilpotent orbits and theta-stable parabolic subalgebras. Representat. Theory 2, 1–32 (1998)MATHCrossRefGoogle Scholar
  14. 14.
    Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. I. J. Algebra 49(2), 525–536 (1977)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. II. J. Algebra 65(2), 373–398 (1980)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tits, J.: Reductive groups over local fields. In: Automorphic Forms, Representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 29–69. Proc. Sympos. Pure Math., XXXIII. American Mathematical Society, Providence (1979)Google Scholar
  17. 17.
    Waldspurger, J.-L.: Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés. Astérisque No. 269 (2001)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

Personalised recommendations