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Nice parabolic subalgebras of reductive Lie algebras


Authors: Karin Baur and Nolan Wallach
Journal: Represent. Theory 9 (2005), 1-29
MSC (2000): Primary 17B45
Published electronically: January 10, 2005
Erratum: Represent. Theory 9 (2005), 267-267.
MathSciNet review: 2123123
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Abstract: This paper gives a classification of parabolic subalgebras of simple Lie algebras over ${\mathbb C}$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three-dimensional simple Lie algebra).


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  • [A] George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR 0557013
  • [B] K. BAUR, A normal form for admissible characters in the sense of Lynch, Represent. Theory 9 (electronic), Amer. Math. Soc. (2005), 30-45.
  • [BHRR] Thomas Brüstle, Lutz Hille, Claus Michael Ringel, and Gerhard Röhrle, The Δ-filtered modules without self-extensions for the Auslander algebra of 𝑘[𝑇]/⟨𝑇ⁿ⟩, Algebr. Represent. Theory 2 (1999), no. 3, 295–312. MR 1715751, 10.1023/A:1009999006899
  • [C] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
  • [CM] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
  • [EK] A.G. ELASHVILI AND V.G. KAC, Classification of good gradings of simple Lie algebras. arXiv:math-ph/0312030v1.
  • [GR] Simon Goodwin and Gerhard Röhrle, Prehomogeneous spaces for parabolic group actions in classical groups, J. Algebra 276 (2004), no. 1, 383–398. MR 2054402, 10.1016/j.jalgebra.2003.11.005
  • [GW] Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
  • [K] Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, 10.1007/BF01390249
  • [L] T. E. LYNCH, Generalized Whittaker vectors and representation theory, Thesis, M.I.T., 1979.
  • [R] R. W. Richardson Jr., Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21–24. MR 0330311
  • [W] N.R. WALLACH, Holomorphic continuation of generalized Jacquet integrals for degenerate principal series, preprint, http://www.math.ucsd.edu/~nwallach/preprints.html

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Additional Information

Karin Baur
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: kbaur@math.ucsd.edu

Nolan Wallach
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: nwallach@math.ucsd.edu

DOI: https://doi.org/10.1090/S1088-4165-05-00262-1
Received by editor(s): October 5, 2004
Received by editor(s) in revised form: November 1, 2004
Published electronically: January 10, 2005
Additional Notes: First named author supported by the Swiss National Science Foundation
Second named author partially supported by an NSF summer grant
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.