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Pieces of nilpotent cones for classical groups


Authors: Pramod N. Achar, Anthony Henderson and Eric Sommers
Journal: Represent. Theory 15 (2011), 584-616
MSC (2010): Primary 17B08, 20G15; Secondary 14L30
DOI: https://doi.org/10.1090/S1088-4165-2011-00393-9
Published electronically: August 22, 2011
MathSciNet review: 2833469
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Abstract: 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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Additional Information

Pramod N. Achar
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisianna 70803-4918
MR Author ID: 701892
Email: pramod@math.lsu.edu

Anthony Henderson
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
MR Author ID: 687061
ORCID: 0000-0002-3965-7259
Email: anthony.henderson@sydney.edu.au

Eric Sommers
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515
Email: esommers@math.umass.edu

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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.