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Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

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Abstract

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik.

The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties.

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References

  1. Andersen, H.-H., Polo, P., Wen, K.X.: Representations of quantum algebras. Invent. Math. 104, 1–59 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arkhipov, S., Bezrukavnikov, R.: Perverse sheaves on affine flags and Langlands dual group. Electronic preprint math.RT/0201073, to appear in Isr. J. Math.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1982)

    MathSciNet  Google Scholar 

  6. Beilinson, A., Bezrukavnikov, R., Mirković, I.: Tilting exersices. Moscow Math. J. 4, 547–557 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Beilinson, A., Ginzburg, V.: Wall-crossing functors and D-modules. Represent. Theory 3, 1–31 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9, 473–527 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bezrukavnikov, R.: Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group. Electronic preprint, math.RT/0201256, to appear in Isr. J. Math.

  10. Bezrukavnikov, R.: Perverse coherent sheaves (after Deligne). Electronic preprint math.AG/0005152

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Bezrukavnikov, R., Mirković, I., Rumynin, D.: Localization of modules for a semisimple Lie algebra in prime characteristic. Electronic preprint, math.RT/0205144, to appear in Ann. Math.

  13. Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53, 1183–1205, 1337 (1989); translation in Math. USSR Izv. 35, 519–541 (1990)

    Google Scholar 

  14. Cline, E., Parshall, B., Scott, L.: Abstract Kazhdan-Lusztig theories. Tohoku Math. J., II. Ser. 45, 511–534 (1993)

    MATH  MathSciNet  Google Scholar 

  15. Demazure, M.: A very simple proof of Bott’s theorem. Invent. Math. 33, 271–272 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. Frenkel, E., Gaitsgory, D.: Local geometric Langlands correspondence and affine Kac-Moody algebras. Preprint math.RT/0508382, 152 pp.

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Ginzburg, V., Kumar, S.: Cohomology of quantum groups at roots of unity. Duke Math. J. 69, 179–198 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goertz, U., Haines, T.J.: The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties. Preprint math.AG/0402143, 71 pp.

  20. Humphreys, J.: Comparing modular representations of semisimple groups and their Lie algebras. Modular interfaces (Riverside, CA 1995), pp. 69–80, AMS/IP Stud. Adv. Math., vol. 4. Providence, RI: Am. Math. Soc. 1997

  21. Lusztig, G.: Introduction to Quantum Groups. Progress in Mathematics, vol. 110. Boston, MA: Birkhäuser 1993

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

    MATH  MathSciNet  Google Scholar 

  23. Ostrik, V.: On the equivariant K-theory of the nilpotent cone. Represent. Theory 4, 296–305 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ostrik, V.: Cohomological supports for quantum groups (russian). Funkts. Anal. Prilozh. 32, 22–34, 95 (1998); translation in Funct. Anal. Appl. 32, 237–246 (1999)

    MATH  MathSciNet  Google Scholar 

  25. Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–223 (1991)

    MathSciNet  Google Scholar 

  26. Soergel, W.: Kategorie \(\mathcal{O}\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3, 421–445 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Roman Bezrukavnikov

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  1. Roman Bezrukavnikov
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Correspondence to Roman Bezrukavnikov.

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To the memory of my father

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Bezrukavnikov, R. Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves. Invent. math. 166, 327–357 (2006). https://doi.org/10.1007/s00222-006-0514-z

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