## Nice parabolic subalgebras of reductive Lie algebras

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- by Karin Baur and Nolan Wallach
- Represent. Theory
**9**(2005), 1-29 - DOI: https://doi.org/10.1090/S1088-4165-05-00262-1
- Published electronically: January 10, 2005
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Erratum: Represent. Theory

**9**(2005), 267-267.

## 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).## References

- George E. Andrews,
*The theory of partitions*, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR**0557013**
[B]B K. Baur, - Thomas Brüstle, Lutz Hille, Claus Michael Ringel, and Gerhard Röhrle,
*The $\Delta$-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. MR**1715751**, DOI 10.1023/A:1009999006899 - 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** - 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]EK A.G. Elashvili and V.G. Kac, - Simon Goodwin and Gerhard Röhrle,
*Prehomogeneous spaces for parabolic group actions in classical groups*, J. Algebra**276**(2004), no. 1, 383–398. MR**2054402**, DOI 10.1016/j.jalgebra.2003.11.005 - 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** - Bertram Kostant,
*On Whittaker vectors and representation theory*, Invent. Math.**48**(1978), no. 2, 101–184. MR**507800**, DOI 10.1007/BF01390249
[L]lT. E. Lynch, - R. W. Richardson Jr.,
*Conjugacy classes in parabolic subgroups of semisimple algebraic groups*, Bull. London Math. Soc.**6**(1974), 21–24. MR**330311**, DOI 10.1112/blms/6.1.21
[W]W N.R. Wallach,

*A normal form for admissible characters in the sense of Lynch*, Represent. Theory 9 (electronic), Amer. Math. Soc. (2005), 30–45.

*Classification of good gradings of simple Lie algebras*. arXiv:math-ph/0312030v1.

*Generalized Whittaker vectors and representation theory*, Thesis, M.I.T., 1979.

*Holomorphic continuation of generalized Jacquet integrals for degenerate principal series*, preprint, http://www.math.ucsd.edu/˜nwallach/preprints.html

## Bibliographic Information

**Karin Baur**- Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
- MR Author ID: 724373
- ORCID: 0000-0002-7665-476X
- Email: kbaur@math.ucsd.edu
**Nolan Wallach**- Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
- MR Author ID: 180225
- Email: nwallach@math.ucsd.edu
- 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 - © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**9**(2005), 1-29 - MSC (2000): Primary 17B45
- DOI: https://doi.org/10.1090/S1088-4165-05-00262-1
- MathSciNet review: 2123123