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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
MathSciNet review: 2123124
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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.


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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

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