Author(s):
Pramod
N.
Achar;
Anthony
Henderson;
Eric
Sommers
Journal:
Represent. Theory
15
(2011),
584-616.
MSC (2010):
Primary 17B08, 20G15;
Secondary 14L30
Posted:
August 22, 2011
Retrieve article in:
PDF
Abstract:
We compare orbits in the nilpotent cone of type , that of type , and Kato's exotic nilpotent cone. We prove that the number of -points in each nilpotent orbit of type or equals that in a corresponding union of orbits, called a type- or type- piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result where corresponding special pieces in types and have the same number of -points. The proof requires studying the case of characteristic , where more direct connections between the three nilpotent cones can be established. We also prove that the type- and type- pieces of the exotic nilpotent cone are smooth in any characteristic.
D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
M. Geck and G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs, New Series, vol. 21, Oxford University Press, New York, 2000. MR 1778802 (2002k:20017)
H. Kraft and C. Procesi, A special decomposition of the nilpotent cone of a classical Lie algebra, Astérisque, no. 173-174 (1989), 271-279. MR 1021514 (90m:17025)
-, Unipotent elements in small characteristic III, with an appendix by G. Lusztig and T. Xue, J. Algebra 329 (2011), 163-189. doi:10.1016/j.jalgebra.2009.12.008, MR 2769321
I. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford Mathematical Monographs, Oxford University Press, 1995. MR 1354144 (96h:05207)
T. Shoji, On the Springer representations of the Weyl groups of classical algebraic groups, Comm. Algebra 7 (1979), no. 16, 1713-1745. MR 546195 (81h:20007a)
-, Green functions attached to limit symbols, in Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, 443-467. MR 2074601 (2005f:05169)
T. Xue, Nilpotent orbits in classical Lie algebras over finite fields of characteristic and the Springer correspondence, Represent. Theory 13 (2009), 371-390. MR 2540701
Pramod
N.
Achar
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisianna 70803-4918
Email:
pramod@math.lsu.edu
Anthony
Henderson
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
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
DOI:
10.1090/S1088-4165-2011-00393-9
PII:
S 1088-4165(2011)00393-9
Received by editor(s):
January 24, 2010
Received by editor(s) in revised form:
June 30, 2010
Posted:
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 of article:
Copyright
2011,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
, that of type 
, and Kato's exotic nilpotent cone. We prove that the number of 
-points in each nilpotent orbit of type 
or 
equals that in a corresponding union of orbits, called a type-
or type-
piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result where corresponding special pieces in types 
and 
have the same number of 
-points. The proof requires studying the case of characteristic 
, where more direct connections between the three nilpotent cones can be established. We also prove that the type-
and type-
pieces of the exotic nilpotent cone are smooth in any characteristic. 
and the Springer correspondence, Represent. Theory 13 (2009), 371-390. 
, arXiv:0911.1350v1. 