Nice parabolic subalgebras of reductive Lie algebras

Authors:
Karin Baur and Nolan Wallach

Journal:
Represent. Theory **9** (2005), 1-29

MSC (2000):
Primary 17B45

DOI:
https://doi.org/10.1090/S1088-4165-05-00262-1

Published electronically:
January 10, 2005

Erratum:
Represent. Theory 9 (2005), 267-267.

MathSciNet review:
2123123

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a classification of parabolic subalgebras of simple Lie algebras over 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).

**[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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/10.1112/blms/6.1.21**[W]**N.R. WALLACH,*Holomorphic continuation of generalized Jacquet integrals for degenerate principal series*, preprint, http://www.math.ucsd.edu/~nwallach/preprints.html

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
17B45

Retrieve articles in all journals with MSC (2000): 17B45

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.