Remarks on Springer’s representations

Lusztig, G.

doi:10.1090/S1088-4165-09-00358-6

Remarks on Springer's representations

Author: G. Lusztig
Journal: Represent. Theory 13 (2009), 391-400
MSC (2000): Primary 20G99
DOI: https://doi.org/10.1090/S1088-4165-09-00358-6
Published electronically: September 3, 2009
MathSciNet review: 2540702
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Abstract: We give an explicit description of a set of irreducible representations of a Weyl group which parametrizes the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group.

[A] D. Alvis, Induce/restrict matrices for exceptional Weyl groups, arxiv:RT/0506377.
[HS] D. F. Holt and N. Spaltenstein, Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic, J. Austral. Math. Soc. Ser. A 38 (1985), no. 3, 330–350. MR 779199
[L1] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125–175. MR 0463275, https://doi.org/10.1007/BF01390002
[L2] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169–178. MR 641425, https://doi.org/10.1016/0001-8708(81)90038-4
[L3] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
[L4] G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, https://doi.org/10.1007/BF01388564
[L5] G. Lusztig, Unipotent elements in small characteristic, Transform. Groups 10 (2005), no. 3-4, 449–487. MR 2183120, https://doi.org/10.1007/s00031-005-0405-1
[L6] G. Lusztig, Unipotent classes and special Weyl group representations, J. Algebra 321 (2009), no. 11, 3418–3449. MR 2510055, https://doi.org/10.1016/j.jalgebra.2008.04.004
[LS] 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
[Se] J.-P. Serre, Letters to G.Lusztig, Nov. 15, 2006, Nov. 9, 2008.
[S1] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
[S2] Nicolas Spaltenstein, Nilpotent classes and sheets of Lie algebras in bad characteristic, Math. Z. 181 (1982), no. 1, 31–48. MR 671712, https://doi.org/10.1007/BF01214979
[S3] N. Spaltenstein, Nilpotent classes in Lie algebras of type 𝐹₄ over fields of characteristic 2, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 517–524. MR 731515
[Sp] T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 0442103, https://doi.org/10.1007/BF01390009
[X1] Ting Xue, Nilpotent orbits in classical Lie algebras over 𝐹_{2ⁿ} and the Springer correspondence, Proc. Natl. Acad. Sci. USA 105 (2008), no. 4, 1126–1128. MR 2375447, https://doi.org/10.1073/pnas.0709626104
[X2] T. Xue, Nilpotent orbits in classical Lie algebras over finite fields of characteristic and the Springer correspondence, Represent. Theory 13 (electronic), (2009), 371-390.