Advertisement

On the nilradical of a parabolic subgroup

  • Karin BaurEmail author
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

We present various approaches to understanding the structure of the nilradical of parabolic subgroups in type A. In particular, we consider the complement of the open dense orbit and describe its irreducible components.

Keywords

Nilradical Parabolic subgroups 

Mathematics Subject Classification 2010:

17B45 

References

  1. 1.
    K. Baur, Richardson elements for classical Lie algebras. J. Algebra 297 (2006), no. 1, 168–185.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    K. Baur, A normal form for admissible characters in the sense of Lynch, Represent. Theory 9 (2005), 30–45.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    K. Baur, K. Erdmann, A. Parker, Δ-filtered modules and nilpotent orbits of a parabolic subgroup in O N. J. Pure Appl. Algebra 215 (2011), no. 5, 885–901.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    K. Baur, S. Goodwin, Richardson elements for parabolic subgroups of classical groups in positive characteristic. Algebr. Represent. Theory 11 (2008), no. 3, 275–297.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    K. Baur, L. Hille, On the complement of the Richardson orbit Math. Z. 272 (2012), no. 1–2, 31–49.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    K. Baur, N. Wallach, Nice Parabolic Subalgebras of Reductive Lie Algebras, Represent. Theory 9 (2005), 1–29.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    K. Baur, N. Wallach, A class of gradings of simple Lie algebras. In Lie algebras, vertex operator algebras and their applications, 3–15, Contemp. Math., 442, AMS, Providence, RI, 2007.Google Scholar
  8. 8.
    T. Brüstle, L. Hille, C. M. Ringel, G. Röhrle, The Δ-filtered modules without self-extensions for the Auslander algebra of \(k[T]/\langle T^{n}\rangle\), Algebr. Represent. Theory 2 (1999), no. 3, 295–312.Google Scholar
  9. 9.
    D.H. Collingwood, W.M. McGovern, Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. xiv+186 pp.Google Scholar
  10. 10.
    W. Hesselink, Polarizations in the classical groups, Math. Zeitschrift 160 (1978), 217–234.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    L. Hille, Aktionen algebraischer Gruppen, geometrische Quotienten und Köcher, Habilitationsschrift, Hamburg 2003.Google Scholar
  12. 12.
    L. Hille, G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1) (1999), 35–52.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    H. Kraft, C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), no. 3, 227–247.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    T. E. Lynch, Generalized Whittaker vectors and representation theory, Thesis, M.I.T., 1979.Google Scholar
  15. 15.
    W.M. McGovern, The adjoint representation and the adjoint action, in Encyclopedia of Mathematical Sciences, vol. 131, Invariant Theory and Algebraic Transformation Groups subseries, Springer, 2002, 159–238.Google Scholar
  16. 16.
    R.W. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bulletin London Math. Society 6 (1974), 21–24.CrossRefzbMATHGoogle Scholar
  17. 17.
    N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics 946, Springer-Verlag, Berlin-New York, 1982.Google Scholar
  18. 18.
    N. Wallach, Holomorphic continuation of generalized Jacquet integrals for degenerate principal series. Represent. Theory 10 (2006), 380–398.CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    H. Yamashita, Multiplicity one theorems for generalized Gel’fand-Graev representations of semisimple Lie groups and Whittaker models for discrete series, Adv. Stud. Pure Math. 14, Academic Press Boston, 1988, 31–121.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria

Personalised recommendations