A normal form for admissible characters in the sense of Lynch

Author:
Karin Baur

Journal:
Represent. Theory **9** (2005), 30-45

MSC (2000):
Primary 17B45; Secondary 17B10

DOI:
https://doi.org/10.1090/S1088-4165-05-00265-7

Published electronically:
January 10, 2005

Erratum:
Represent. Theory **9** (2005), 525-525.

MathSciNet review:
2123124

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Parabolic subalgebras $\mathfrak {p}$ of semisimple Lie algebras define a $\mathbb {Z}$-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of $\mathfrak {g}$ on which the adjoint group of $\mathfrak {p}$ acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of $\mathfrak {g}_1$ is exactly one that involves all simple root spaces. It is, however, difficult to write down such nilpotent elements for general parabolic subalgebras. In this paper we give an explicit construction of admissible elements in $\mathfrak {g}_1$ that uses as few root spaces as possible.

- 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 https://doi.org/10.1023/A%3A1009999006899
[BW]bw K. Baur, N. R. Wallach, - 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** - 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 https://doi.org/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 https://doi.org/10.1007/BF01390249
[L]l T.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 https://doi.org/10.1112/blms/6.1.21
[W]wa N.R. Wallach,

*Nice parabolic subalgebras of reductive Lie algebras*, Represent. Theory 9 (electronic), Amer. Math. Soc. (2005), 1–29.

*Generalized Whittaker vectors and representation theory*, Ph.D. Thesis, 1979, MIT Libraries (http://theses.mit.edu)

*Holomorphic continuation of generalized Jacquet integrals for degenerate principal series*, in preparation.

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
17B45,
17B10

Retrieve articles in all journals with MSC (2000): 17B45, 17B10

Additional 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

Keywords:
Parabolic subalgebras,
admissible characters

Received by editor(s):
October 5, 2004

Received by editor(s) in revised form:
November 22, 2004

Published electronically:
January 10, 2005

Additional Notes:
The author was supported by a DARPA grant and by Uarda Frutiger-Fonds (Freie Akademische Stiftung)

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.