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Algebraic representations and constructible sheaves

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Abstract

I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p > 0 and Lusztig’s character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg–Mirković conjecture) in the spirit of the Kazhdan–Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GL n ) which should eventually make computations more feasible.

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References

  1. P.N. Achar, S. Makisumi, S. Riche and G. Williamson, Free-monodromic mixed tilting sheaves on flag varieties, preprint, arXiv:1703.05843.

  2. P.N. Achar, S. Makisumi, S. Riche and G. Williamson, Koszul duality for Kac–Moody groups and characters of tilting modules, preprint, arXiv:1706.00183.

  3. P.N. Achar and S. Riche, Modular perverse sheaves on flag varieties. III: positivity conditions, preprint, arXiv:1408.4189.

  4. Achar, P.N.; Rider, L.: The affine Grassmannian and the Springer resolution in positive characteristic. Compos. Math. 152, 2627–2677 (2016)

    Article  MathSciNet  Google Scholar 

  5. Achar, P.N.; Rider, L.: Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture. Acta Math. 215, 183–216 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. P.N. Achar and S. Riche, Modular perverse sheaves on flag varieties. I: tilting and parity sheaves. With a joint appendix with Geordie Williamson, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 325–370.

  7. Achar, P.N.; Riche, S.: Modular perverse sheaves on flag varieties. II: Koszul duality and formality. Duke Math. J. 165, 161–215 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.N. Achar and S. Riche, Reductive groups, the loop Grassmannian, and the Springer resolution, preprint, arXiv:1602.04412.

  9. H.H. Andersen, Modular representations of algebraic groups, In: The Arcata Conference on Representations of Finite Groups, Arcata, CA, 1986, Proc. Sympos. Pure Math., 47, Amer. Math. Soc., Providence, RI, 1987, pp. 23–36.

  10. H.H. Andersen, Filtrations and tilting modules, Ann. Sci. École Norm. Sup. (4), 30 (1997), 353–366.

  11. H.H. Andersen, Tilting modules for algebraic groups, In: Algebraic Groups and Their Representations, Cambridge, 1997, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 25–42.

  12. H.H. Andersen, A sum formula for tilting filtrations, In: Commutative Algebra, Homological Algebra and Representation Theory, Catania, Genoa, Rome, 1998, J. Pure Appl. Algebra, 152, Elsevier, Amsterdam, 2000, pp. 17–40.

  13. H.H. Andersen, Tilting modules for algebraic and quantum groups, In: Algebra—Representation Theory, Constanta, 2000, NATO Sci. Ser. II Math. Phys. Chem., 28, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–21.

  14. Andersen, H.H.; Jantzen, J.C.; Soergel, W.: Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque, 220. Soc. Math, France (1994)

    MATH  Google Scholar 

  15. Andersen, H.H.; Kulkarni, U.: Sum formulas for reductive algebraic groups. Adv. Math. 217, 419–447 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Angiono, A quantum version of the algebra of distributions of \(\mathrm{SL}_2\), preprint, arXiv:1607.04869.

  17. Arkhipov, S.; Bezrukavnikov, R.: Perverse sheaves on affine flags and Langlands dual group. With an appendix by Bezrukavrikov and Ivan Mirković. Israel J. Math. 170, 135–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Bernstein, Algebraic D-modules, Lecture notes (unpublished).

  20. Bezrukavnikov, R.: On two geometric realizations of an affine Hecke algebra. Publ. Math. Inst. Hautes Études Sci. 123, 1–67 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Bezrukavnikov and I. Mirković, Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution, Ann. of Math. (2), 178 (2013), 835–919.

  22. Bezrukavnikov, R.; Mirković, I.; Rumynin, D.: Singular localization and intertwining functors for reductive Lie algebras in prime characteristic. Nagoya Math. J. 184, 1–55 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Bezrukavnikov, I. Mirković and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic. With an appendix by Bezrukavnikov and Simon Riche, Ann. of Math. (2), 167 (2008), 945–991.

  24. R. Bezrukavnikov and S. Riche, Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. Éc. Norm. Supér. (4), 45 (2012), 535–599.

  25. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968.

  26. Brundan, J.: On the definition of Kac-Moody 2-category. Math. Ann. 364, 353–372 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Brylinski, R.K.: Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc. 2, 517–533 (1989)

    MathSciNet  MATH  Google Scholar 

  28. Chuang, J.: Derived equivalence in \(\mathrm{SL}_2(p^2)\). Trans. Amer. Math. Soc. 353, 2897–2913 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and \(\mathfrak{s}\mathfrak{l}_2\)-categorification, Ann. of Math. (2), 167 (2008), 245–298.

  30. Cline, E.; Parshall, B.; Scott, L.; van der Kallen, W.: Rational and generic cohomology. Invent. Math. 39, 143–163 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  31. Curtis, C.W.: Representations of Lie algebras of classical type with applications to linear groups. J. Math. Mech. 9, 307–326 (1960)

    MathSciNet  MATH  Google Scholar 

  32. Deodhar, V.V.: On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials. J. Algebra 111, 483–506 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Donkin, S.: The blocks of a semisimple algebraic group. J. Algebra 67, 36–53 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Donkin, Rational Representations of Algebraic Groups. Tensor Products and Filtration, Lecture Notes in Math., 1140, Springer-Verlag, 1985.

  35. Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212, 39–60 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Donkin, An introduction to the Lusztig conjecture, In: Representations of Reductive Groups, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1998, pp. 173–187.

  37. Elias, B.; Williamson, G.: Soergel calculus. Represent. Theory 20, 295–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. B. Elias and G. Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2), 180 (2014), 1089–1136.

  39. K. Erdmann, Symmetric groups and quasi-hereditary algebras, In: Finite Dimensional Algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of Algebras and Related Topics, Ottawa, Canada, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 123–161.

  40. Fiebig, P.: Lusztig’s conjecture as a moment graph problem. Bull. Lond. Math. Soc. 42, 957–972 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Fiebig, P.: The multiplicity one case of Lusztig’s conjecture. Duke Math. J. 153, 551–571 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Fiebig, P.: Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture. J. Amer. Math. Soc. 24, 133–181 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fiebig, P.: An upper bound on the exceptional characteristics for Lusztig’s character formula. J. Reine Angew. Math. 673, 1–31 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Fiebig, P.; Williamson, G.: Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. Ann. Inst. Fourier (Grenoble) 64, 489–536 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Finkelberg and I. Mirković, Semi-infinite flags. I. Case of global curve P 1, In: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999, pp. 81–112.

  46. W.J. Haboush, Central differential operators on split semisimple groups over fields of positive characteristic, In: Séminaire d’Algèbre Paul Dubreil et Marie–Paule Malliavin, 32ème année, Paris, 1979, Lecture Notes in Math., 795, Springer-Verlag, 1980, pp. 35–85.

  47. X. He and G. Williamson, Soergel calculus and Schubert calculus, preprint, arXiv:1502.04914; Bull. Inst. Math. Acad. Sin. (N.S.), to appear.

  48. T.L. Hodge, P. Karuppuchamy and L.L. Scott, Remarks on the ABG induction theorem, preprint, arXiv:1603.05699.

  49. J.E. Humphreys, Modular Representations of Finite Groups of Lie Type, London Math. Soc. Lecture Note Ser., 326, Cambridge Univ. Press, Cambridge, 2006.

  50. Iwahori, N.; Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of \({\mathfrak{p}}\)-adic Chevalley groups. Inst. Hautes Études Sci. Publ. Math. 25, 5–48 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  51. Jantzen, J.C.: Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren. Math. Z. 140, 127–149 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  52. Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und kontravariante Formen. J. Reine Angew. Math. 290, 117–141 (1977)

    MathSciNet  MATH  Google Scholar 

  53. J.C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., 750, Springer-Verlag, 1979.

  54. J.C. Jantzen, Modular representations of reductive groups, In: Group Theory, Beijing 1984, Lecture Notes in Math., 1185, Springer-Verlag, 1986, pp. 118–154.

  55. J.C. Jantzen, Representations of Algebraic Groups. Second ed., Math. Surveys Monogr., 107, Amer. Math. Soc., Providence, RI, 2003.

  56. J.C. Jantzen, Character formulae from Hermann Weyl to the present, In: Groups and Analysis, London Math. Soc. Lecture Note Ser., 354, Cambridge Univ. Press, Cambridge, 2008, pp. 232–270.

  57. Jensen, J.G.: On the character of some modular indecomposable tilting modules for \(\mathrm{SL}_3\). J. Algebra 232, 397–419 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Jensen and G. Williamson, The p-canonical basis for Hecke algebras. Perspectives in categorification, to appear (2015).

  59. D. Juteau, Modular representations of reductive groups and geometry of affine Grassmannians, preprint, arXiv:0804.2041.

  60. Juteau, D.: Decomposition numbers for perverse sheaves. Ann. Inst. Fourier (Grenoble) 59, 1177–1229 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  61. Juteau, D.; Mautner, C.; Williamson, G.: Parity sheaves. J. Amer. Math. Soc. 27, 1169–1212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  62. D. Juteau, C. Mautner and G. Williamson, Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 257–275.

  63. Kaneda, M.: Based modules and good filtrations in algebraic groups. Hiroshima Math. J. 28, 337–344 (1998)

    MathSciNet  MATH  Google Scholar 

  64. Kashiwara, M.; Tanisaki, T.: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. Duke Math. J. 77, 21–62 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kashiwara, M.; Tanisaki, T.: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case. Duke Math. J. 84, 771–813 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  66. Kato, S.: On the Kazhdan-Lusztig polynomials for affine Weyl groups. Adv. in Math. 55, 103–130 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  67. Kazhdan, D.; Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  68. D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, In: Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979, Proc. Sympos. Pure Math., 36, Amer. Math. Soc., Providence, RI, 1980, pp. 185–203.

  69. Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. I, II. J. Amer. Math. Soc. 6(905–947), 949–1011 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  70. Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. III. J. Amer. Math. Soc. 7, 335–381 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  71. Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. IV. J. Amer. Math. Soc. 7, 383–453 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  72. Khovanov, M.; Lauda, A.D.: A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13, 309–347 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  73. Khovanov, M.; Lauda, A.D.: A diagrammatic approach to categorification of quantum groups. II. Trans. Amer. Math. Soc. 363, 2685–2700 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  74. Libedinsky, N.: Light leaves and Lusztig’s conjecture. Adv. Math. 280, 772–807 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. Littelmann, P.: Good filtrations and decomposition rules for representations with standard monomial theory. J. Reine Angew. Math. 433, 161–180 (1992)

    MathSciNet  MATH  Google Scholar 

  76. G. Lusztig, Some problems in the representation theory of finite Chevalley groups, In: The Santa Cruz Conference on Finite Groups, Univ. California, Santa Cruz, CA, 1979, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, pp. 313–317.

  77. G. Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, In: Analysis and Topology on Singular Spaces. II, III, Luminy, 1981, Astérisque, 101, Soc. Math. France, Paris, 1983, pp. 208–229.

  78. Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra. J. Amer. Math. Soc. 3, 257–296 (1990)

    MathSciNet  MATH  Google Scholar 

  79. Lusztig, G.: On quantum groups. J. Algebra 131, 466–475 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  80. Lusztig, G.: Monodromic systems on affine flag manifolds. Proc. Roy. Soc. London Ser. A 445(1923), 231–246 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  81. G. Lusztig, Errata: Monodromic systems on affine flag manifolds, Proc. Roy. Soc. London Ser. A, 445, no. 1923 (1994), 231–246, Proc. Roy. Soc. London Ser. A, 450, no. 1940 (1995), 731–732.

  82. Lusztig, G.: On the character of certain irreducible modular representations. Represent. Theory 19, 3–8 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  83. G. Lusztig and G. Williamson, On the character of certain tilting modules, preprint, arXiv:1502.04904.

  84. O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup. (4), 23 (1990), 625–644.

  85. O. Mathieu, Tilting modules and their applications, In: Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama–Kyoto, 1997, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000, pp. 145–212.

  86. C. Mautner and S. Riche, Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic–Vilonen conjecture, preprint, arXiv:1501.07369; J. Eur. Math. Soc. (JEMS), to appear.

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

  88. Nadler, D.: Perverse sheaves on real loop Grassmannians. Invent. Math. 159, 1–73 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  89. T. Okuyama, Derived equivalence in \(sl(2, q)\), preprint, 2000.

  90. B.J. Parshall, Cohomology of algebraic groups, In: The Arcata Conference on Representations of Finite Groups, Arcata, CA, 1986, Proc. Sympos. Pure Math., 47, Amer. Math. Soc., Providence, RI, 1987, pp. 233–248.

  91. J. Paradowski, Filtrations of modules over the quantum algebra, In: Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Univ. Park, PA, 1991, Proc. Sympos. Pure Math., 56, Amer. Math. Soc., Providence, RI, 1994, pp. 93–108.

  92. A. Parker, Some remarks on a result of Jensen and tilting modules for \(SL_3(k)\) and \(q-GL_3(k)\), preprint, arXiv:0809.2249.

  93. Polo, P.: Modules associés aux variétés de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 308, 123–126 (1989)

    MathSciNet  MATH  Google Scholar 

  94. S. Riche and G. Williamson, Tilting modules and the p-canonical basis, preprint, arXiv:1512.08296.

  95. 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)

    Article  MathSciNet  MATH  Google Scholar 

  96. R. Rouquier, 2-Kac–Moody algebras, preprint, arXiv:0812.5023.

  97. L. Scott, Linear and nonlinear group actions, and the Newton Institute program, In: Algebraic Groups and Their Representations, Cambridge, 1997, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 1–23.

  98. Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren. Represent. Theory 1, 115–132 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  99. Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1, 83–114 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  100. Soergel, W.: On the relation between intersection cohomology and representation theory in positive characteristic. J. Pure Appl. Algebra 152, 311–335 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  101. T.A. Springer, Quelques applications de la cohomologie d’intersection, In: Séminaire Bourbaki. Vol. 1981/1982, Astérisque, 92, Soc. Math. France, Paris, 1982, pp.249–273.

  102. D.-N. Verma, The rôle of affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras, In: Lie Groups and Their Representations, Halsted, New York, 1975, pp. 653–705.

  103. Wang, J.P.: Sheaf cohomology on \(G/B\) and tensor products of Weyl modules. J. Algebra 77, 162–185 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  104. G. Williamson, A reducible characteristic variety in type A, In: Representations of Reductive Groups, Progr. Math., 312, Birkhäuser/Springer, Cham, 2015, pp. 517–532.

  105. G. Williamson, Local Hodge theory of Soergel bimodules, Acta Math., to appear (2016).

  106. Williamson, G.: On torsion in the intersection cohomology of Schubert varieties. J. Algebra 475, 207–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  107. G. Williamson, Schubert calculus and torsion explosion, preprint, arXiv:1309.5055v2; J. Amer. Math. Soc., to appear.

  108. Williamson, G.; Braden, T.: Modular intersection cohomology complexes on flag varieties. Math. Z. 272, 697–727 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia

    Geordie Williamson

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Communicated by: Hiraku Nakajima

This article is based on the 18th Takagi Lectures that the author delivered at the University of Tokyo on November 5–6, 2016.

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Williamson, G. Algebraic representations and constructible sheaves. Jpn. J. Math. 12, 211–259 (2017). https://doi.org/10.1007/s11537-017-1646-1

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