From conjugacy classes in the Weyl group to unipotent classes

Lusztig, G.

doi:10.1090/S1088-4165-2011-00396-4

From conjugacy classes in the Weyl group to unipotent classes

Author: G. Lusztig
Journal: Represent. Theory 15 (2011), 494-530
MSC (2010): Primary 20G99
DOI: https://doi.org/10.1090/S1088-4165-2011-00396-4
Published electronically: June 8, 2011
MathSciNet review: 2833465
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes in .

[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
[EG] Erich W. Ellers and Nikolai Gordeev, Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math. 214 (2004), no. 2, 245–261. MR 2042932, https://doi.org/10.2140/pjm.2004.214.245
[Ge] Meinolf Geck, On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. 1 (1996), No. 15, 293–317. MR 1418951
[Ch] M.Geck, G.Hiss, F.Lübeck, G.Malle and G.Pfeiffer, A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175-210.
[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
[Ka] Noriaki Kawanaka, Unipotent elements and characters of finite Chevalley groups, Osaka Math. J. 12 (1975), no. 2, 523–554. MR 384914
[KL] D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), no. 2, 129–168. MR 947819, https://doi.org/10.1007/BF02787119
[L1] G.Lusztig, On the reflection representation of a finite Chevalley group, Representation theory of Lie groups, LMS Lect. Notes Ser. 34, Cambridge Univ. Press, 1979, pp. 325-337.
[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] George Lusztig, Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355–408. MR 1043271, https://doi.org/10.2307/1971496
[L5] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442
[L6] G.Lusztig, On some partitions of a flag manifold, arxiv:0906.1505.
[Lü] F.Lübeck, http://www.math.rwth-aachen.de/~Frank.Luebeck /chev/Green/.
[Mi] Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups 𝐸₇ and 𝐸₈, Tokyo J. Math. 3 (1980), no. 2, 391–461. MR 605099, https://doi.org/10.3836/tjm/1270473003
[Sh] Toshiaki Shoji, Character sheaves and almost characters of reductive groups. I, II, Adv. Math. 111 (1995), no. 2, 244–313, 314–354. MR 1318530, https://doi.org/10.1006/aima.1995.1024
[Sp1] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
[Sp2] G. Lusztig and N. Spaltenstein, On the generalized Springer correspondence for classical groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 289–316. MR 803339, https://doi.org/10.2969/aspm/00610289
[Sp3] N. Spaltenstein, Polynomials over local fields, nilpotent orbits and conjugacy classes in Weyl groups, Astérisque 168 (1988), 10–11, 191–217. Orbites unipotentes et représentations, I. MR 1021497
[Sp4] N. Spaltenstein, On the Kazhdan-Lusztig map for exceptional Lie algebras, Adv. Math. 83 (1990), no. 1, 48–74. MR 1069387, https://doi.org/10.1016/0001-8708(90)90068-X
[St] Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80. MR 180554