Koszul duality for Kac–Moody groups and characters of tilting modules

Achar, Pramod¹ ; Makisumi, Shotaro² ; Riche, Simon³ ; Williamson, Geordie⁴

¹ Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
² Department of Mathematics, Columbia University, New York, New York 10027
³ Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
⁴ School of Mathematics and Statistics F07, University of Sydney NSW 2006, Australia

Journal of the American Mathematical Society, Tome 32 (2019) no. 1, pp. 261-310

Résumé

We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic $\ell$ in terms of $\ell$-Kazhdan–Lusztig polynomials, for $\ell > h$ the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if $\ell \ge 2h-2$. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.

Détail

DOI : 10.1090/jams/905

Bibliographie

[1] Achar, Pramod N., Riche, Simon Koszul duality and semisimplicity of Frobenius Ann. Inst. Fourier (Grenoble) 2013 1511 1612

[2] Achar, Pramod N., Riche, Simon Modular perverse sheaves on flag varieties I: tilting and parity sheaves Ann. Sci. Éc. Norm. Supér. (4) 2016 325 370

[3] Achar, Pramod N., Riche, Simon Modular perverse sheaves on flag varieties, II: Koszul duality and formality Duke Math. J. 2016 161 215

[4] Achar, Pramod N., Rider, Laura Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture Acta Math. 2015 183 216

[5] Achar, Pramod N., Rider, Laura The affine Grassmannian and the Springer resolution in positive characteristic Compos. Math. 2016 2627 2677

[6] 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 1994 321

[7] Arkhipov, Sergey, Bezrukavnikov, Roman Perverse sheaves on affine flags and Langlands dual group Israel J. Math. 2009 135 183

[8] Arkhipov, Sergey, Bezrukavnikov, Roman, Ginzburg, Victor Quantum groups, the loop Grassmannian, and the Springer resolution J. Amer. Math. Soc. 2004 595 678

[9] Beä­Linson, A. A. On the derived category of perverse sheaves 1987 27 41

[10] Beä­Linson, Alexandre, Bernstein, Joseph Localisation de 𝑔-modules C. R. Acad. Sci. Paris Sér. I Math. 1981 15 18

[11] Beilinson, A., Bezrukavnikov, R., Mirkoviä‡, I. Tilting exercises Mosc. Math. J. 2004

[12] Beilinson, Alexander, Ginzburg, Victor Wall-crossing functors and 𝒟-modules Represent. Theory 1999 1 31

[13] Beilinson, Alexander, Ginzburg, Victor, Soergel, Wolfgang Koszul duality patterns in representation theory J. Amer. Math. Soc. 1996 473 527

[14] Bezrukavnikov, Roman Cohomology of tilting modules over quantum groups and 𝑡-structures on derived categories of coherent sheaves Invent. Math. 2006 327 357

[15] Bezrukavnikov, Roman, Yun, Zhiwei On Koszul duality for Kac-Moody groups Represent. Theory 2013 1 98

[16] Elias, Ben, Williamson, Geordie Soergel calculus Represent. Theory 2016 295 374

[17] Fiebig, Peter An upper bound on the exceptional characteristics for Lusztig’s character formula J. Reine Angew. Math. 2012 1 31

[18] Ginsburg, Victor Perverse sheaves and 𝐶*-actions J. Amer. Math. Soc. 1991 483 490

[19] Humphreys, J. E. Comparing modular representations of semisimple groups and their Lie algebras 1997 69 80

[20] Jantzen, Jens Carsten Character formulae from Hermann Weyl to the present 2008 232 270

[21] Jensen, Lars Thorge, Williamson, Geordie The 𝑝-canonical basis for Hecke algebras 2017 333 361

[22] Juteau, Daniel, Mautner, Carl, Williamson, Geordie Parity sheaves J. Amer. Math. Soc. 2014 1169 1212

[23] Kashiwara, Masaki, Tanisaki, Toshiyuki Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case Duke Math. J. 1996 771 813

[24] Kazhdan, David, Lusztig, George Schubert varieties and Poincaré duality 1980 185 203

[25] Kazhdan, D., Lusztig, G. Tensor structures arising from affine Lie algebras. IV J. Amer. Math. Soc. 1994 383 453

[26] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory 2002

[27] Libedinsky, Nicolas Sur la catégorie des bimodules de Soergel J. Algebra 2008 2675 2694

[28] Lusztig, George Some problems in the representation theory of finite Chevalley groups 1980 313 317

[29] Lusztig, George Errata: “Monodromic systems on affine flag manifolds” [Proc. Roy. Soc. London Ser. A 445 (1994), no. 1923, 231–246 Proc. Roy. Soc. London Ser. A 1995 731 732

[30] Lusztig, George, Williamson, Geordie Billiards and tilting characters for 𝑆𝐿₃ SIGMA Symmetry Integrability Geom. Methods Appl. 2018

[31] Mathieu, Olivier Formules de caractères pour les algèbres de Kac-Moody générales Astérisque 1988 267

[32] Mathieu, Olivier Construction d’un groupe de Kac-Moody et applications Compositio Math. 1989 37 60

[33] Ostrik, V. Tensor ideals in the category of tilting modules Transform. Groups 1997 279 287

[34] Rider, Laura Formality for the nilpotent cone and a derived Springer correspondence Adv. Math. 2013 208 236

[35] Soergel, Wolfgang Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe J. Amer. Math. Soc. 1990 421 445

[36] Soergel, Wolfgang Character formulas for tilting modules over quantum groups at roots of one 1999 161 172

[37] Soergel, Wolfgang Character formulas for tilting modules over Kac-Moody algebras Represent. Theory 1998 432 448

[38] Soergel, Wolfgang Langlands’ philosophy and Koszul duality 2001 379 414

[39] Soergel, Wolfgang On the relation between intersection cohomology and representation theory in positive characteristic J. Pure Appl. Algebra 2000 311 335

[40] Soergel, Wolfgang Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen J. Inst. Math. Jussieu 2007 501 525

[41] Springer, T. A. Quelques applications de la cohomologie d’intersection 1982 249 273

[42] Tits, Jacques Groupes associés aux algèbres de Kac-Moody Astérisque 1989

[43] Williamson, Geordie, Braden, Tom Modular intersection cohomology complexes on flag varieties Math. Z. 2012 697 727

[44] Williamson, Geordie Schubert calculus and torsion explosion J. Amer. Math. Soc. 2017 1023 1046

[45] Yun, Zhiwei Weights of mixed tilting sheaves and geometric Ringel duality Selecta Math. (N.S.) 2009 299 320