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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Pieces of nilpotent cones for classical groups
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by Pramod N. Achar, Anthony Henderson and Eric Sommers
Represent. Theory 15 (2011), 584-616
Published electronically: August 22, 2011


We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato’s exotic nilpotent cone. We prove that the number of $\mathbb {F}_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig’s result where corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\mathbb {F}_q$-points. The proof requires studying the case of characteristic $2$, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.
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Bibliographic Information
  • Pramod N. Achar
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisianna 70803-4918
  • MR Author ID: 701892
  • Email:
  • Anthony Henderson
  • Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
  • MR Author ID: 687061
  • ORCID: 0000-0002-3965-7259
  • Email:
  • Eric Sommers
  • Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515
  • Email:
  • Received by editor(s): January 24, 2010
  • Received by editor(s) in revised form: June 30, 2010
  • Published electronically: August 22, 2011
  • Additional Notes: The first author’s research was supported by Louisiana Board of Regents grant NSF(2008)-LINK-35 and by National Security Agency grant H98230-09-1-0024.
    The second author’s research was supported by Australian Research Council grant DP0985184.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 15 (2011), 584-616
  • MSC (2010): Primary 17B08, 20G15; Secondary 14L30
  • DOI:
  • MathSciNet review: 2833469