Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some remarks on Richardson orbits in complex symmetric spaces

Author: Alfred G. Noël
Journal: Math. Comp. 75 (2006), 395-417
MSC (2000): Primary 17B05, 17B10, 17B20, 22E30
Published electronically: September 29, 2005
MathSciNet review: 2176406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Roger W. Richardson proved that any parabolic subgroup of a complex semisimple Lie group admits an open dense orbit in the nilradical of its corresponding parabolic subalgebra. In the case of complex symmetric spaces we show that there exist some large classes of parabolic subgroups for which the analogous statement which fails in general, is true. Our main contribution is the extension of a theorem of Peter E. Trapa (in 2005) to real semisimple exceptional Lie groups.

References [Enhancements On Off] (What's this?)

  • 1. Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. MR 1013112
  • 2. Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
  • 3. G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), 41-52. MR 0527733 (82g:20070)
  • 4. A. G. Noël, Computing theta-stable parabolic subalgebras using LiE, Lecture Notes Comput. Sci., Springer-Verlag 3039 (2004), 335-342.
  • 5. R. W. Richardson Jr., Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21–24. MR 0330311,
  • 6. J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39, No. 1 (1987), 127-138. MR 0867991 (88g:53053)
  • 7. Patrice Tauvel, Quelques résultats sur les algèbres de Lie symétriques, Bull. Sci. Math. 125 (2001), no. 8, 641–665 (French, with French summary). MR 1872599,
  • 8. Peter E. Trapa, Richardson orbits for real classical groups, J. Algebra 286 (2005), no. 2, 361–385. MR 2128022,
  • 9. È. B. Vinberg, The classification of nilpotent elements of graded Lie algebras, Dokl. Akad. Nauk SSSR 225 (1975), no. 4, 745–748 (Russian). MR 0506488
  • 10. M. A. A. Van Leeuwen, A. M. Cohen, and B. Lisser, LiE A package for Lie Group Computations, Computer Algebra Nederland, Amsterdam, The Netherlands (1992).
  • 11. D. Djokovic Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1987) 577-585. MR 0926619 (89b:17010)
  • 12. D. Djokovic, Classification of nilpotent elements in simple real Lie algebras $ E_{6(6)}$ and $ E_{6(-26)}$ and description of their centralizers, J. Algebra 116 (1988) 196-207. MR 0944155 (89k:17022)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 17B05, 17B10, 17B20, 22E30

Retrieve articles in all journals with MSC (2000): 17B05, 17B10, 17B20, 22E30

Additional Information

Alfred G. Noël
Affiliation: Mathematics Department, The University of Massachusetts, Boston, Massachusetts 02125-3393

Keywords: Parabolic group, nilpotent orbits, prehomogeneous spaces
Received by editor(s): March 15, 2004
Published electronically: September 29, 2005
Additional Notes: The author was partially supported by an NSF research opportunity award sponsored by David Vogan of MIT. He thanks him for the support. The author is also grateful to Donald R. King and Peter E. Trapa for several discussions about the content of this paper. Finally, he expresses his thanks to the referee for his kind words.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.