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

Lusztig, G.

doi:10.1090/S1088-4165-2012-00422-8

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an affine algebraic group over an algebraically closed field whose identity component $G^{0}$ is reductive. Let be the Weyl group of and let be a connected component of whose image in $G/G^{0}$ is unipotent. In this paper we define a map from the set of ``twisted conjugacy classes'' in to the set of unipotent $G^{0}$ -conjugacy classes in generalizing an earlier construction which applied when is connected.

[Ca] R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
[DL] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, https://doi.org/10.2307/1971021
[GP] Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
[GKP] Meinolf Geck, Sungsoon Kim, and Götz Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2000), no. 2, 570–600. MR 1769289, https://doi.org/10.1006/jabr.1999.8253
[H1] Xuhua He, Minimal length elements in some double cosets of Coxeter groups, Adv. Math. 215 (2007), no. 2, 469–503. MR 2355597, https://doi.org/10.1016/j.aim.2007.04.005
[H2] Xuhua He, On the affineness of Deligne-Lusztig varieties, J. Algebra 320 (2008), no. 3, 1207–1219. MR 2427638, https://doi.org/10.1016/j.jalgebra.2007.12.028
[HL] Xuhua He and George Lusztig, A generalization of Steinberg’s cross section, J. Amer. Math. Soc. 25 (2012), no. 3, 739–757. MR 2904572, https://doi.org/10.1090/S0894-0347-2012-00728-0
[HN] X. He and S. Nie, Minimal length elements of finite Coxeter groups, arxiv:1108.0282.
[L1] George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8–12, 1977. MR 518617
[L2] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
[L3] George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226–265, 266–315. MR 806210, https://doi.org/10.1016/0001-8708(85)90064-7
[L4] G. Lusztig, Character sheaves on disconnected groups. I, Represent. Theory 7 (2003), 374–403. MR 2017063, https://doi.org/10.1090/S1088-4165-03-00204-8
[L5] G. Lusztig, Character sheaves on disconnected groups. I, Represent. Theory 7 (2003), 374–403. MR 2017063, https://doi.org/10.1090/S1088-4165-03-00204-8
[L6] G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, Represent. Theory 15 (2011), 494–530. MR 2833465, https://doi.org/10.1090/S1088-4165-2011-00396-4
[L7] G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes, II, Represent. Theory 16 (2012), 189–211. MR 2904567, https://doi.org/10.1090/S1088-4165-2012-00411-3
[L8] G. Lusztig, Elliptic elements in a Weyl group: a homogeneity property, Represent. Theory 16 (2012), 127–151. MR 2888173, https://doi.org/10.1090/S1088-4165-2012-00409-5
[L9] G. Lusztig, On certain varieties attached to a Weyl group element, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 4, 377–414. MR 2907958
[LX] George Lusztig and Ting Xue, Elliptic Weyl group elements and unipotent isometries with 𝑝=2, Represent. Theory 16 (2012), 270–275. MR 2915753, https://doi.org/10.1090/S1088-4165-2012-00415-0
[M1] Gunter Malle, Generalized Deligne-Lusztig characters, J. Algebra 159 (1993), no. 1, 64–97. MR 1231204, https://doi.org/10.1006/jabr.1993.1147
[M2] Gunter Malle, Green functions for groups of types 𝐸₆ and 𝐹₄ in characteristic 2, Comm. Algebra 21 (1993), no. 3, 747–798. MR 1204754, https://doi.org/10.1080/00927879308824595
[M3] G. Malle, Personal communication, May 2011.
[Sp] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610