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Representation Theory

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Nilpotent orbits and theta-stable parabolic subalgebras

Author: Alfred G. Noël
Journal: Represent. Theory 2 (1998), 1-32
MSC (1991): Primary 17B20, 17B70
Published electronically: February 3, 1998
MathSciNet review: 1600330
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Abstract: In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra ${\mathfrak {g}}$ under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on ${\mathfrak {p}}_{{}_{\mathbb {C}}}$, the “complex symmetric space associated with ${\mathfrak {g}}$”. The Kostant-Sekiguchi correspondence, a bijection between nilpotent orbits in ${\mathfrak {g}}$ and nilpotent orbits in ${\mathfrak {p}}_{{}_{\mathbb {C}}}$, is also used. We identify a fundamental set of noticed nilpotents in ${\mathfrak {p}}_{{}_{\mathbb {C}}}$ and show that they allow us to recover all other nilpotents. Finally, we study the behaviour of a principal orbit, that is an orbit of maximal dimension, under our classification. This is not done in the other classification schemes currently available in the literature.

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Additional Information

Alfred G. Noël
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts, 02115; Peritus Software Services Inc. 304 Concord Road, Billerica, Massachusetts 01821

Keywords: Parabolic subalgebras, nilpotent orbits, reductive Lie algebras
Received by editor(s): August 11, 1997
Received by editor(s) in revised form: December 3, 1997
Published electronically: February 3, 1998
Additional Notes: The author thanks his advisor, Donald R. King, for his helpful suggestions.
Article copyright: © Copyright 1998 American Mathematical Society