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On two geometric realizations of an affine Hecke algebra

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Abstract

The article is a contribution to the local theory of geometric Langlands duality. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra associated to a reductive group \(G\) and Grothendieck group of equivariant coherent sheaves on Steinberg variety of Langlands dual group \({G\check {\ }}\); this isomorphism due to Kazhdan–Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures.

The paper is a continuation of the author’s joint work with Arkhipov, it relies on the technical material developed in a joint work with Yun.

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References

  1. S. Arkhipov and R. Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, Isr. J. Math., 170 (2009), 135–184.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Arkhipov, R. Bezrukavnikov and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Am. Math. Soc., 17 (2004), 595–678.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Arkhipov and D. Gaitsgory, Another realization of the category of modules over the small quantum group, Adv. Math., 173 (2003), 114–143.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Arinkin and R. Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J., 10 (2010), 3–29.

    MathSciNet  MATH  Google Scholar 

  5. D. Arinkin and D. Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture, Sel. Math. New Ser., 21 (2015), 1–199.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Beilinson, How to glue perverse sheaves, in K-Theory, Arithmetic and Geometry, Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987.

    Google Scholar 

  7. A. Beilinson and J. Bernstein, A generalization of Casselman’s submodule theorem, in Representation Theory of Reductive Groups, Progr. Math., vol. 40, Park City, Utah, 1982, pp. 35–52, Birkhäuser Boston, Boston, 1983.

    Chapter  Google Scholar 

  8. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I, Astérisque, vol. 100, Luminy, 1981, 1982.

    Google Scholar 

  9. A. Beilinson, R. Bezrukavnikov and I. Mirković, Tilting exercises, Mosc. Math. J., 4 (2004), 547–557.

    MathSciNet  MATH  Google Scholar 

  10. D. Ben-Zvi and D. Nadler, The character theory of a complex group. arXiv:0904.1247, preprint.

  11. J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, Lecture Notes in Mathematics, vol. 1578, Springer, Berlin, 1994.

    MATH  Google Scholar 

  12. R. Bezrukavnikov, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, Isr. J. Math., 170 (2009), 185–206.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution, in Proceeding of the International Congress of Mathematicians, vol. 2, Madrid, Spain, pp. 1119–1144, 2006.

    Google Scholar 

  14. R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and \(t\)-structures on derived categories of coherent sheaves, Invent. Math., 166 (2006), 327–357.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory, 7 (2003), 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Bezrukavnikov, A. Braverman and I. Mirković, Some results about the geometric Whittaker model, Adv. Math., 186 (2004), 143–152.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Bezrukavnikov and M. Finkelberg, Equivariant Satake category and Kostant–Whittaker reduction, Mosc. Math. J., 8 (2008), 39–72.

    MathSciNet  MATH  Google Scholar 

  18. R. Bezrukavnikov and A. Lachowska, The small quantum group and the Springer resolution, in Quantum Groups, Contemp. Math., vol. 433, pp. 89–101, Am. Math. Soc., Providence, 2007.

    Chapter  Google Scholar 

  19. R. Bezrukavnikov and Q. Lin, Highest weight modules at the critical level and noncommutative Springer resolution, Contemp. Math., 565 (2012), 15–27.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Bezrukavnikov and I. Mirković, Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution, Ann. Math., 178 (2013), 835–919, with an Appendix by E. Sommers.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Bezrukavnikov and S. Riche, Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. Éc. Norm. Super., 45 (2012), 535–599.

    MathSciNet  MATH  Google Scholar 

  22. R. Bezrukavnikov and S. Riche, Hodge \(D\)-modules and braid group actions, in preparation.

  23. R. Bezrukavnikov, Z. Yun, On Koszul duality for Kac-Moody groups, Represent. Theory, 17 (2013), 1–98, with Appendices by Z. Yun.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser Boston, Boston, 1997.

    MATH  Google Scholar 

  25. P. Deligne, La conjecture de Weil. II, Publ. Math. IHES, 52 (1980), 137–252.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Deligne, Catégories tannakiennes in The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, pp. 111–195, Birkhäuser Boston, Boston, 1990.

    Google Scholar 

  27. E. Frenkel, D. Gaitsgory and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. Math., 153 (2001), 699–748.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Frenkel and D. Gaitsgory, D-modules on the affine flag variety and representations of affine Kac-Moody algebras, Represent. Theory, 13 (2009), 477–608.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Gaitsgory, Appendix: braiding compatibilities, in Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, pp. 91–100, Math. Soc. Japan, Tokyo, 2004.

    Google Scholar 

  31. D. Gaitsgory, The notion of category over an algebraic stack. arXiv:math/0507192, preprint.

  32. D. Gaitsgory, Sheaves of categories and the notion of 1-affineness. arXiv:1306.4304, preprint.

  33. S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, in Current Developments in Mathematics, vol. 2006 pp. 35–180, International Press, Somerville, 2008.

    Google Scholar 

  34. A. Grothendieck, Éléments de géométrie algébrique, III, Publ. Math. IHES, 11 (1961) (partie 1).

  35. D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 87 (1987), 153–215.

    Article  MathSciNet  MATH  Google Scholar 

  36. G. Lusztig, Cells in affine Weyl groups, in Algebraic Groups and Related Topics, Advanced Studies in Pure Math., vol. 6, pp. 255–287, Kinokuniya/North Holland, Tokyo/Amsterdam, 1985.

    Google Scholar 

  37. G. Lusztig, Cells in affine Weyl groups. IV, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 36 (1989), 297–328.

    MathSciNet  MATH  Google Scholar 

  38. G. Lusztig, Singularities, character formulas and a \(q\)-analogue of weight multiplicities, Astérisque, 101–102 (1983), 208–229.

    MathSciNet  MATH  Google Scholar 

  39. G. Lusztig, Some examples of square integrable representations of semisimple \(p\)-adic groups, Trans. Am. Math. Soc., 277 (1983), 623–653.

    MathSciNet  MATH  Google Scholar 

  40. G. Lusztig, Equivariant K-theory and representations of Hecke algebras, Proc. Am. Math. Soc., 94 (1985), 337–342.

    MathSciNet  MATH  Google Scholar 

  41. I. Mirković and S. Riche, Linear Koszul duality, Compos. Math., 146 (2010), 233–258.

    Article  MathSciNet  MATH  Google Scholar 

  42. I. Mirković and S. Riche, Linear Koszul duality and affine Hecke algebras. arXiv:0903.0678, preprint.

  43. A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Am. Math. Soc., 9 (1996), 205–236.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Neeman, The connection between the \(K\)-theory localization theorem of Thomasn, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Super., 25 (1992), 547–566.

    MathSciNet  MATH  Google Scholar 

  45. M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Seconde partie, Invent. Math., 13 (1971), 1–89.

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Thomason and T. Trobaugh, Higher algebraic \(K\)-theory of schemes and of derived categories, in The Grothendieck Festschrift, vol. 3, pp. 247–435, Birkhäuser, Basel, 1990.

    Chapter  Google Scholar 

  47. J.-L. Verdier, Spécialisation de faisceaux et monodromie modérée, Astérisque, 101–102 (1983), 332–364.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts ave., Cambridge, MA, 02139, USA

    Roman Bezrukavnikov

  2. International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, 101000, Russia

    Roman Bezrukavnikov

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  1. Roman Bezrukavnikov
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Correspondence to Roman Bezrukavnikov.

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Bezrukavnikov, R. On two geometric realizations of an affine Hecke algebra. Publ.math.IHES 123, 1–67 (2016). https://doi.org/10.1007/s10240-015-0077-x

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