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From conjugacy classes in the Weyl group to unipotent classes, III


Author: G. Lusztig
Journal: Represent. Theory 16 (2012), 450-488
MSC (2010): Primary 20G99
DOI: https://doi.org/10.1090/S1088-4165-2012-00422-8
Published electronically: September 7, 2012
MathSciNet review: 2968566
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Abstract: Let $ G$ be an affine algebraic group over an algebraically closed field whose identity component $ G^{0}$ is reductive. Let $ W$ be the Weyl group of $ G$ and let $ D$ be a connected component of $ G$ whose image in $ G/G^{0}$ is unipotent. In this paper we define a map from the set of ``twisted conjugacy classes'' in $ W$ to the set of unipotent $ G^{0}$-conjugacy classes in $ D$ generalizing an earlier construction which applied when $ G$ is connected.


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  • [Ca] R.W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1-59. MR 0318337 (47:6884)
  • [DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. Math. 103 (1976), 103-161. MR 0393266 (52:14076)
  • [GP] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Clarendon Press Oxford, 2000. MR 1778802 (2002k:20017)
  • [GKP] M. Geck, S. Kim and G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2000), 570-600. MR 1769289 (2001h:20049)
  • [H1] X. He, Minimal length elements in some double cosets of Coxeter groups, Adv. Math. 215 (2007), 469-503. MR 2355597 (2009g:20088)
  • [H2] X. He, On the affineness of the Deligne-Lusztig varieties, J. Algebra 320 (2008), 1207-1219. MR 2427638 (2009c:20085)
  • [HL] X. He and G. Lusztig, A generalization of Steinberg's cross section, Jour. Amer. Math. Soc. 25 (2012), 739-757. MR 2904572
  • [HN] X. He and S. Nie, Minimal length elements of finite Coxeter groups, arxiv:1108.0282.
  • [L1] G. Lusztig, Representations of finite Chevalley groups, Regional Conf. Series in Math., vol. 39, Amer. Math. Soc., 1978. MR 518617 (80f:20045)
  • [L2] G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton Univ. Press, 1984. MR 742472 (86j:20038)
  • [L3] G. Lusztig, Character sheaves IV, Adv. Math. 59 (1986), 1-63. MR 825086 (87m:20118b)
  • [L4] G. Lusztig, Character sheaves on disconnected groups I, Represent.Theory 7 (2003), 374-403. MR 2017063 (2006d:20090a)
  • [L5] G. Lusztig, Character sheaves on disconnected groups II, Represent. Theory 8 (2004), 72-124. MR 2048588 (2006d:20090b)
  • [L6] G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, Represent. Theory 15 (2011), 494-530. MR 2833465 (2012g:20092)
  • [L7] G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, II, Represent. Theory 16 (2012), 189-211. MR 2904567
  • [L8] G. Lusztig, Elliptic elements in a Weyl group: a homogeneity property, Represent. Theory 16 (2012), 127-151. MR 2888173
  • [L9] G. Lusztig, On certain varieties attached to a Weyl group element, Bull. Math. Inst. Acad. Sinica (N.S.) 6 (2011), no. 4, 377-414. MR 2907958
  • [LX] G. Lusztig and T. Xue, Elliptic Weyl group elements and unipotent isometries with $ p=2$, Represent. Theory 16 (2012), 270-275. MR 2915753
  • [M1] G. Malle, Generalized Deligne-Lusztig characters, J. Algebra 159 (1993), 64-97. MR 1231204 (94i:20025)
  • [M2] G. Malle, Green functions for groups of type $ E_{6}$ and $ F_{4}$ in characteristic $ 2$, Commun. Algebra 21 (1993), 747-798. MR 1204754 (94c:20077)
  • [M3] G. Malle, Personal communication, May 2011.
  • [Sp] N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., vol. 946, Springer-Verlag, 1982. MR 672610 (84a:14024)

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Additional Information

G. Lusztig
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S1088-4165-2012-00422-8
Received by editor(s): October 13, 2011
Received by editor(s) in revised form: May 11, 2012
Published electronically: September 7, 2012
Additional Notes: Supported in part by the National Science Foundation
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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