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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Nilpotent orbits and theta-stable parabolic subalgebras
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by Alfred G. Noël PDF
Represent. Theory 2 (1998), 1-32 Request permission


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
  • Email:,
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Represent. Theory 2 (1998), 1-32
  • MSC (1991): Primary 17B20, 17B70
  • DOI:
  • MathSciNet review: 1600330