Advertisement

Japanese Journal of Mathematics

, Volume 12, Issue 2, pp 211–259 | Cite as

Algebraic representations and constructible sheaves

  • Geordie Williamson
Special Feature: The Takagi Lectures
  • 159 Downloads

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.

Keywords and phrases

algebraic groups representations perverse sheaves Hecke category 

Mathematics Subject Classification (2010)

17B10 20G15 14F05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AMRW17a.
    P.N. Achar, S. Makisumi, S. Riche and G. Williamson, Free-monodromic mixed tilting sheaves on flag varieties, preprint, arXiv:1703.05843.
  2. AMRW17b.
    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. AR14a.
    P.N. Achar and S. Riche, Modular perverse sheaves on flag varieties. III: positivity conditions, preprint, arXiv:1408.4189.
  4. AR14b.
    Achar, P.N.; Rider, L.: The affine Grassmannian and the Springer resolution in positive characteristic. Compos. Math. 152, 2627–2677 (2016)MathSciNetCrossRefGoogle Scholar
  5. AR15.
    Achar, P.N.; Rider, L.: Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture. Acta Math. 215, 183–216 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. AR16a.
    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.Google Scholar
  7. AR16b.
    Achar, P.N.; Riche, S.: Modular perverse sheaves on flag varieties. II: Koszul duality and formality. Duke Math. J. 165, 161–215 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. AR16c.
    P.N. Achar and S. Riche, Reductive groups, the loop Grassmannian, and the Springer resolution, preprint, arXiv:1602.04412.
  9. And87.
    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.Google Scholar
  10. And97.
    H.H. Andersen, Filtrations and tilting modules, Ann. Sci. École Norm. Sup. (4), 30 (1997), 353–366.Google Scholar
  11. And98.
    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.Google Scholar
  12. And00.
    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.Google Scholar
  13. And01.
    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.Google Scholar
  14. AJS94.
    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)zbMATHGoogle Scholar
  15. AK08.
    Andersen, H.H.; Kulkarni, U.: Sum formulas for reductive algebraic groups. Adv. Math. 217, 419–447 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Ang16.
    I. Angiono, A quantum version of the algebra of distributions of \(\mathrm{SL}_2\), preprint, arXiv:1607.04869.
  17. AB09.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. ABG04.
    Arkhipov, S.; Bezrukavnikov, R.; Ginzburg, V.: Quantum groups, the loop Grassmannian, and the Springer resolution. J. Amer. Math. Soc. 17, 595–678 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ber.
    J. Bernstein, Algebraic D-modules, Lecture notes (unpublished).Google Scholar
  20. Bez16.
    Bezrukavnikov, R.: On two geometric realizations of an affine Hecke algebra. Publ. Math. Inst. Hautes Études Sci. 123, 1–67 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. BM13.
    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.Google Scholar
  22. BMR06.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. BMR08.
    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.Google Scholar
  24. BR13.
    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.Google Scholar
  25. Bou68.
    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.Google Scholar
  26. Bru16.
    Brundan, J.: On the definition of Kac-Moody 2-category. Math. Ann. 364, 353–372 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Bry89.
    Brylinski, R.K.: Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits. J. Amer. Math. Soc. 2, 517–533 (1989)MathSciNetzbMATHGoogle Scholar
  28. Chu01.
    Chuang, J.: Derived equivalence in \(\mathrm{SL}_2(p^2)\). Trans. Amer. Math. Soc. 353, 2897–2913 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. CR08.
    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.Google Scholar
  30. CPSvdK77.
    Cline, E.; Parshall, B.; Scott, L.; van der Kallen, W.: Rational and generic cohomology. Invent. Math. 39, 143–163 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Cur60.
    Curtis, C.W.: Representations of Lie algebras of classical type with applications to linear groups. J. Math. Mech. 9, 307–326 (1960)MathSciNetzbMATHGoogle Scholar
  32. Deo87.
    Deodhar, V.V.: On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials. J. Algebra 111, 483–506 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Don80.
    Donkin, S.: The blocks of a semisimple algebraic group. J. Algebra 67, 36–53 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Don85.
    S. Donkin, Rational Representations of Algebraic Groups. Tensor Products and Filtration, Lecture Notes in Math., 1140, Springer-Verlag, 1985.Google Scholar
  35. Don93.
    Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212, 39–60 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Don98.
    S. Donkin, An introduction to the Lusztig conjecture, In: Representations of Reductive Groups, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1998, pp. 173–187.Google Scholar
  37. EW13.
    Elias, B.; Williamson, G.: Soergel calculus. Represent. Theory 20, 295–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  38. EW14.
    B. Elias and G. Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2), 180 (2014), 1089–1136.Google Scholar
  39. Erd94.
    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.Google Scholar
  40. Fie10a.
    Fiebig, P.: Lusztig’s conjecture as a moment graph problem. Bull. Lond. Math. Soc. 42, 957–972 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Fie10b.
    Fiebig, P.: The multiplicity one case of Lusztig’s conjecture. Duke Math. J. 153, 551–571 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  42. Fie11.
    Fiebig, P.: Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture. J. Amer. Math. Soc. 24, 133–181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Fie12.
    Fiebig, P.: An upper bound on the exceptional characteristics for Lusztig’s character formula. J. Reine Angew. Math. 673, 1–31 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  44. FW14.
    Fiebig, P.; Williamson, G.: Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. Ann. Inst. Fourier (Grenoble) 64, 489–536 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. FM99.
    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.Google Scholar
  46. Hab80.
    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.Google Scholar
  47. HW15.
    X. He and G. Williamson, Soergel calculus and Schubert calculus, preprint, arXiv:1502.04914; Bull. Inst. Math. Acad. Sin. (N.S.), to appear.
  48. HKS16.
    T.L. Hodge, P. Karuppuchamy and L.L. Scott, Remarks on the ABG induction theorem, preprint, arXiv:1603.05699.
  49. Hum06.
    J.E. Humphreys, Modular Representations of Finite Groups of Lie Type, London Math. Soc. Lecture Note Ser., 326, Cambridge Univ. Press, Cambridge, 2006.Google Scholar
  50. IM65.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  51. Jan74.
    Jantzen, J.C.: Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren. Math. Z. 140, 127–149 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  52. Jan77.
    Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und kontravariante Formen. J. Reine Angew. Math. 290, 117–141 (1977)MathSciNetzbMATHGoogle Scholar
  53. Jan79.
    J.C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., 750, Springer-Verlag, 1979.Google Scholar
  54. Jan86.
    J.C. Jantzen, Modular representations of reductive groups, In: Group Theory, Beijing 1984, Lecture Notes in Math., 1185, Springer-Verlag, 1986, pp. 118–154.Google Scholar
  55. Jan03.
    J.C. Jantzen, Representations of Algebraic Groups. Second ed., Math. Surveys Monogr., 107, Amer. Math. Soc., Providence, RI, 2003.Google Scholar
  56. Jan08.
    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.Google Scholar
  57. Jen00.
    Jensen, J.G.: On the character of some modular indecomposable tilting modules for \(\mathrm{SL}_3\). J. Algebra 232, 397–419 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  58. JW15.
    T. Jensen and G. Williamson, The p-canonical basis for Hecke algebras. Perspectives in categorification, to appear (2015).Google Scholar
  59. Jut08.
    D. Juteau, Modular representations of reductive groups and geometry of affine Grassmannians, preprint, arXiv:0804.2041.
  60. Jut09.
    Juteau, D.: Decomposition numbers for perverse sheaves. Ann. Inst. Fourier (Grenoble) 59, 1177–1229 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  61. JMW14.
    Juteau, D.; Mautner, C.; Williamson, G.: Parity sheaves. J. Amer. Math. Soc. 27, 1169–1212 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  62. JMW16.
    D. Juteau, C. Mautner and G. Williamson, Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér. (4), 49 (2016), 257–275.Google Scholar
  63. Kan98.
    Kaneda, M.: Based modules and good filtrations in algebraic groups. Hiroshima Math. J. 28, 337–344 (1998)MathSciNetzbMATHGoogle Scholar
  64. KT95.
    Kashiwara, M.; Tanisaki, T.: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. Duke Math. J. 77, 21–62 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  65. KT96.
    Kashiwara, M.; Tanisaki, T.: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case. Duke Math. J. 84, 771–813 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  66. Kat85.
    Kato, S.: On the Kazhdan-Lusztig polynomials for affine Weyl groups. Adv. in Math. 55, 103–130 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  67. KL79.
    Kazhdan, D.; Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53, 165–184 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  68. KL80.
    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.Google Scholar
  69. KL93.
    Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. I, II. J. Amer. Math. Soc. 6(905–947), 949–1011 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  70. KL94a.
    Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. III. J. Amer. Math. Soc. 7, 335–381 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  71. KL94b.
    Kazhdan, D.; Lusztig, G.: Tensor structures arising from affine Lie algebras. IV. J. Amer. Math. Soc. 7, 383–453 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  72. KL09.
    Khovanov, M.; Lauda, A.D.: A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13, 309–347 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  73. KL11.
    Khovanov, M.; Lauda, A.D.: A diagrammatic approach to categorification of quantum groups. II. Trans. Amer. Math. Soc. 363, 2685–2700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  74. Lib15.
    Libedinsky, N.: Light leaves and Lusztig’s conjecture. Adv. Math. 280, 772–807 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  75. Lit92.
    Littelmann, P.: Good filtrations and decomposition rules for representations with standard monomial theory. J. Reine Angew. Math. 433, 161–180 (1992)MathSciNetzbMATHGoogle Scholar
  76. Lus80.
    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.Google Scholar
  77. Lus83.
    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.Google Scholar
  78. Lus90a.
    Lusztig, G.: Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra. J. Amer. Math. Soc. 3, 257–296 (1990)MathSciNetzbMATHGoogle Scholar
  79. Lus90b.
    Lusztig, G.: On quantum groups. J. Algebra 131, 466–475 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  80. Lus94.
    Lusztig, G.: Monodromic systems on affine flag manifolds. Proc. Roy. Soc. London Ser. A 445(1923), 231–246 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  81. Lus95.
    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.Google Scholar
  82. Lus15.
    Lusztig, G.: On the character of certain irreducible modular representations. Represent. Theory 19, 3–8 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  83. LW15.
    G. Lusztig and G. Williamson, On the character of certain tilting modules, preprint, arXiv:1502.04904.
  84. Mat90.
    O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup. (4), 23 (1990), 625–644.Google Scholar
  85. Mat00.
    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.Google Scholar
  86. MR13.
    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. MV07.
    I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2), 166 (2007), 95–143.Google Scholar
  88. Nad05.
    Nadler, D.: Perverse sheaves on real loop Grassmannians. Invent. Math. 159, 1–73 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  89. Oku00.
    T. Okuyama, Derived equivalence in \(sl(2, q)\), preprint, 2000.Google Scholar
  90. Par87.
    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.Google Scholar
  91. Par94.
    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.Google Scholar
  92. Par08.
    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. Pol89.
    Polo, P.: Modules associés aux variétés de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 308, 123–126 (1989)MathSciNetzbMATHGoogle Scholar
  94. RW15.
    S. Riche and G. Williamson, Tilting modules and the p-canonical basis, preprint, arXiv:1512.08296.
  95. Rin91.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  96. Rou08.
    R. Rouquier, 2-Kac–Moody algebras, preprint, arXiv:0812.5023.
  97. Sco98.
    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.Google Scholar
  98. Soe97a.
    Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren. Represent. Theory 1, 115–132 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  99. Soe97b.
    Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1, 83–114 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  100. Soe00.
    Soergel, W.: On the relation between intersection cohomology and representation theory in positive characteristic. J. Pure Appl. Algebra 152, 311–335 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  101. Spr82.
    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.Google Scholar
  102. Ver75.
    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.Google Scholar
  103. Wan82.
    Wang, J.P.: Sheaf cohomology on \(G/B\) and tensor products of Weyl modules. J. Algebra 77, 162–185 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  104. Wil15.
    G. Williamson, A reducible characteristic variety in type A, In: Representations of Reductive Groups, Progr. Math., 312, Birkhäuser/Springer, Cham, 2015, pp. 517–532.Google Scholar
  105. Wil16a.
    G. Williamson, Local Hodge theory of Soergel bimodules, Acta Math., to appear (2016).Google Scholar
  106. Wil16b.
    Williamson, G.: On torsion in the intersection cohomology of Schubert varieties. J. Algebra 475, 207–228 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  107. Wil16c.
    G. Williamson, Schubert calculus and torsion explosion, preprint, arXiv:1309.5055v2; J. Amer. Math. Soc., to appear.
  108. WB12.
    Williamson, G.; Braden, T.: Modular intersection cohomology complexes on flag varieties. Math. Z. 272, 697–727 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan KK 2017

Authors and Affiliations

  1. 1.School of Mathematics and Statistics F07University of SydneySydneyAustralia

Personalised recommendations