Selecta Mathematica

, Volume 23, Issue 1, pp 203–243 | Cite as

Constructible sheaves on nilpotent cones in rather good characteristic

  • Pramod N. Achar
  • Anthony Henderson
  • Daniel Juteau
  • Simon Riche
Article
  • 121 Downloads

Abstract

We study some aspects of modular generalized Springer theory for a complex reductive group G with coefficients in a field \(\Bbbk \) under the assumption that the characteristic \(\ell \) of \(\Bbbk \) is rather good for G, i.e. \(\ell \) is good and does not divide the order of the component group of the centre of G. We prove a comparison theorem relating the characteristic-\(\ell \) generalized Springer correspondence to the characteristic-0 version. We also consider Mautner’s characteristic-\(\ell \) ‘cleanness conjecture’; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.

Mathematics Subject Classification

Primary 17B08 20G05 

Notes

Acknowledgments

We thank Carl Mautner for discussions concerning his conjectures, which motivated some of the results in the second half of the paper. We used the development version [19] of the GAP Chevie package [11].

References

  1. 1.
    Achar, P., Henderson, A., Juteau, D., Riche, S.: Modular generalized Springer correspondence I: the general linear group. J. Eur. Math. Soc. Preprint arXiv:1307.2702
  2. 2.
    Achar, P., Henderson, A., Juteau, D., Riche, S.: Modular generalized Springer correspondence II: classical groups. J. Eur. Math. Soc. Preprint arXiv:1404.1096
  3. 3.
    Achar, P., Henderson, A., Juteau, D., Riche, S.: Modular generalized Springer correspondence III: exceptional groups. Preprint arXiv:1507.00401
  4. 4.
    Achar, P., Henderson, A., Juteau, D., Riche, S.: Weyl group actions on the Springer sheaf. Proc. Lond. Math. Soc. 108(6), 1501–1528 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beĭlinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers, Analyse et topologie sur les espaces singuliers, I (Luminy,1981), Astérisque. Soc. Math. Paris, France 100, 5–171 (1982)Google Scholar
  6. 6.
    Beĭlinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9, 473–527 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bernstein, J., Lunts, V.: Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer (1994)Google Scholar
  8. 8.
    Bonnafé, C.: Actions of relative Weyl groups. I. J. Group Theory 7(1), 1–37 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Co., New York (1993)MATHGoogle Scholar
  10. 10.
    Curtis, C.W., Reiner, I.: Methods of Representation Theory, vol. II. Wiley, New York (1987)MATHGoogle Scholar
  11. 11.
    Geck, M., Hiss, G., Luebeck, F., Malle, G., Pfeiffer, G.: CHEVIE: a system for computing and processing generic character tables. Appl. Algebra Eng. Comm. Comput. 7, 175–210 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Juteau, D., Lecouvey, C., Sorlin, K.: Springer basic sets and modular Springer correspondence for classical types. Preprint arXiv:1410.1477
  13. 13.
    Juteau, D.: Modular Springer correspondence, decomposition matrices and basic sets. Preprint arXiv:1410.1471
  14. 14.
    Lusztig, G.: Intersection cohomology complexes on a reductive group. Invent. Math. 75, 205–272 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lusztig, G.: Character sheaves II. Adv. Math. 57(3), 226–265 (1985)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lusztig, G.: Character sheaves V. Adv. Math. 61(2), 103–155 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lusztig, G.: Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994), CMS Conf. Proc. Am. Math. Soc. 16, 217–275 (1995)Google Scholar
  18. 18.
    Mautner, C.: A geometric Schur functor. Selecta Math. (N.S.) 20, 961–977 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Michel, J.: The development version of the CHEVIE package of GAP3. J. Algebra 435, 308–336 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mostow, G.D.: Fully reducible subgroups of algebraic groups. Am. J. Math. 78, 200–221 (1956)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Rider, L., Russell, A.: Perverse sheaves on the nilpotent cone and Lusztig’s generalized Springer correspondence. In: Proceedings of Southeastern Lie Theory Workshop Series. Preprint arXiv:1409.7132

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Pramod N. Achar
    • 1
  • Anthony Henderson
    • 2
  • Daniel Juteau
    • 3
  • Simon Riche
    • 4
  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  3. 3.Laboratoire de Mathématiques Nicolas OresmeUniversité de CaenCaen CedexFrance
  4. 4.Université Blaise Pascal - Clermont-Ferrand II, Laboratoire de Mathématiques, CNRS, UMR 6620Aubière CedexFrance

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