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Epipelagic representations and rigid local systems

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Abstract

We construct automorphic representations for quasi-split groups G over the function field \(F=k(t)\) one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified \({}^{L}{G}\)-local systems over \(\mathbb {P}^{1}-\{0,\infty \}\) that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when G is a classical group.

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References

  1. Bezrukavnikov, R.: On tensor categories attached to cells in affine Weyl groups. In: Representation Theory of Algebraic Groups and Quantum Groups. Adv. Stud. Pure Math. vol. 40, pp 69–90. Mathematical Society of Japan, Tokyo (2004)

  2. Carter, R.W.: Finite groups of Lie type. Conjugacy classes and complex characters. Pure and Applied Mathematics (New York). Wiley, New York (1985)

  3. Deligne, P., Milne, J.S.: Tannakian categories. In: Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900. Springer, Berlin (1982)

  4. Dettweiler, M., Reiter, S.: Rigid local systems and motives of type \(G_2\). With an appendix by M. Dettweiler and N. M. Katz. Compos. Math. 146(4), 929–963 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gaitsgory, D.: On de Jongs conjecture. Isr. J. Math. 157, 155–191 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5, 41–68 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Frenkel, E., Gross, B.: A rigid irregular connection on the projective line. Ann. Math. (2) 170(3), 1469–1512 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144(2), 253–280 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gross, B., Levy, P., Reeder, M., Yu, J.-K.: Gradings of positive rank on simple Lie algebras. Transform. Gr. 17(4), 1123–1190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gross, B., Reeder, M.: Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154, 431–508 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heinloth, J., Ngô, B.-C., Yun, Z.: Kloosterman sheaves for reductive groups. Ann. Math. (2) 177(1), 241–310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Katz, N.M.: Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, vol. 116. Princeton University Press, Princeton (1988)

  13. Lafforgue, V.: Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale. arXiv:1209.5352

  14. Lusztig, G., Spaltenstein, N.: Induced unipotent classes. J. Lond. Math. Soc. (2) 19(1), 41–52 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lusztig, G.: A class of irreducible representations of a Weyl group. Proc. Nederl. Akad. Ser. A 82, 323–335 (1979)

    MathSciNet  MATH  Google Scholar 

  16. Lusztig, G.: Cells in affine Weyl groups I. Algebraic groups and related topics (Kyoto/Nagoya, 1983), 255–287 Adv. Stud. Pure Math., 6, North-Holland, Amsterdam (1985)

  17. Lusztig, G.: Cells in affine Weyl groups II. J. Algebra 109(2), 536–548 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lusztig, G.: Cells in affine Weyl groups III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34(2), 223–243 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Lusztig, G.: Cells in affine Weyl groups IV. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(2), 297–328 (1989)

    MathSciNet  MATH  Google Scholar 

  20. Lusztig, G.: Cells in affine Weyl groups and tensor categories. Adv. Math. 129(1), 85–98 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lusztig, G.: From conjugacy classes in the Weyl group to unipotent classes. Represent. Theory 15, 494–530 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lusztig, G.: Unipotent classes and special Weyl group representations. J. Algebra 321(11), 3418–3449 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Macdonald, I.G.: Some irreducible representations of Weyl groups. Bull. Lond. Math. Soc. 4, 148–150 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Reeder, M., Yu, J.-K.: Epipelagic representations and invariant theory. J. AMS 27, 437–477 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Springer, T.A.: Regular elements of finite reflection groups. Invent. Math. 25, 159–198 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vinberg, E.B.: The Weyl group of a graded Lie algebra. Izv. AN SSSR. Ser. Mat. 40(3), 488–526 (1976) (in Russian); English translation: Math. USSR-Izv. bf 10, 463–495 (1976)

  28. Yun, Z.: Global Springer theory. Adv. Math. 228(1), 266–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yun, Z.: Motives with exceptional Galois groups and the inverse Galois problem. Invent. Math. 196, 267–337 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yun, Z.: Rigidity in automorphic representations and local systems. In: Jerison, D., Kisin, M., Mrowka, T., Stanley, R., Yau, H.-T., Yau, S.-T. (eds.) Current Developments in Mathematics 2013, pp. 73–168. International Press, Somerville (2014)

  31. Yun, Z., Zhu, X.: A monodromic geometric Satake equivalence (in preparation)

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Acknowledgments

The author thank B. Gross, M. Reeder and J-K. Yu for inspiring conversations. He also thanks an anonymous referee for useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA

    Zhiwei Yun

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  1. Zhiwei Yun
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Correspondence to Zhiwei Yun.

Additional information

Supported by the NSF Grant DMS-1302071 and the Packard Fellowship.

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Yun, Z. Epipelagic representations and rigid local systems. Sel. Math. New Ser. 22, 1195–1243 (2016). https://doi.org/10.1007/s00029-015-0204-z

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