Modular generalized Springer correspondence III: exceptional groups
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We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type \(G_2\).
Mathematics Subject ClassificationPrimary 17B08 20G05
We are grateful to Jean Michel for implementing many functions on unipotent classes, including the generalized Springer correspondence (with characteristic zero coefficients), in the development version of the GAP3 Chevie package . We would also like to thank the anonymous referee for a careful reading and detailed comments on the paper.
- 2.Achar, P., Henderson, A., Juteau, D., Riche, S.: Modular generalized Springer correspondence II: classical groups. J. Eur. Math. Soc. arXiv:1404.1096 (to appear)
- 12.Green, J.A.: On the indecomposable representations of a finite group. Math. Z. 70 (1958/59), 430–445Google Scholar
- 14.Juteau, D.: Modular Springer correspondence, decomposition matrices and basic sets. Bull. Soc. Math. France arXiv:1410.1471 (to appear)
- 15.Juteau, D., Lecouvey, C., Sorlin, K.: Springer basic sets and modular Springer correspondence for classical types. arXiv:1410.1477
- 22.Lusztig, G.: Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, 217–275Google Scholar
- 23.Lusztig, G.: On the generalized Springer correspondence. arXiv:1608.02223
- 24.Lusztig, G., Spaltenstein, N.: On the generalized Springer correspondencefor classical groups, Algebraic groups and related topics (Kyoto/Nagoya,1983), Adv. Stud. Pure Math., vol. 6, North-Holland Publishing Co., Amsterdam, 1985, pp. 289–316Google Scholar
- 26.Michel, J.: The development version of the CHEVIE package of GAP3. arXiv:1310.7905
- 27.Spaltenstein, N.: On the generalized Springer correspondence for exceptional groups, Algebraic groups and related topics (Kyoto/Nagoya,1983), Adv. Stud. Pure Math., vol. 6, North-Holland Publishing Co., Amsterdam, 1985, pp. 317–338Google Scholar