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Astérisque - Titles - 397 (2018)

Titles < 2018

Tilting modules and the p-canonical basis
Simon Riche, Geordie Williamson
Astérisque 397 (2018), ix+184 pages
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Résumé :
Modules basculants et la base p-canonique
Dans cet ouvrage nous proposons une nouvelle approche à l'étude des modules basculants pour les groupes algébriques réductifs sur des corps de caractéristique positive. Nous conjecturons que les foncteurs de translation induisent une action de la catégorie de Hecke (diagrammatique) du groupe de Weyl affine sur le bloc principal. Cette conjecture implique des formules de caractères pour les modules simples et les modules basculants en termes de la base p-canonique, ainsi qu'une description du bloc principal comme le quotient anti-sphérique de la catégorie de Hecke. Nous démontrons notre conjecture pour le groupe GL_n() en utilisant la théorie des représentations des algèbres 2-Kac-Moody. Enfin, nous prouvons que la catégorie de Hecke diagrammatique d'un groupe de Coxeter cristallographique général peut être décrite en termes de faisceaux à parité sur la variété de drapeaux du groupe de Kac-Moody correspondant.

Mots-clefs : Modules basculants, groupes algébriques, faisceaux à parité, catégorie de Hecke.

Abstract:
In this book we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the antispherical quotient of the Hecke category. We prove our conjecture for GL_n() using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac-Moody group.

Keywords: Tilting modules, algebraic groups, parity sheaves, Hecke category.

Class. math. : 17B10, 20G15, 14F05.


ISBN : 978-2-85629-880-0
ISSN : 0303-1179 (print) 2492-5926 (electronic)
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
2
Achar, Pramod N. and Riche, Simon
Modular perverse sheaves on flag varieties I: tilting and parity sheaves
Ann. Sci. Éc. Norm. Supér. 49 (2016) 325–370
Math Reviews MR3481352
3
Achar, Pramod N. and Riche, Simon
Modular perverse sheaves on flag varieties, II: Koszul duality and formality
Duke Math. J. 165 (2016) 161–215
Math Reviews MR3450745
4
Achar, Pramod N. and Riche, Simon
Modular perverse sheaves on flag varieties III: positivity conditions
Trans. Amer. Math. Soc. 370 (2018) 447–485
5
6
Andersen, Henning Haahr
Tilting modules for algebraic groups
in Algebraic groups and their representations (Cambridge, 1997)
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 517 (1998) 25–42
Math Reviews MR1670762
7
Andersen, Henning Haahr
A sum formula for tilting filtrations
J. Pure Appl. Algebra 152 (2000) 17–40
Math Reviews MR1783982
8
Andersen, Henning Haahr and Jantzen, Jens Carsten and Soergel, Wolfgang
Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p
Astérisque 220 (1994)
Math Reviews MR1272539
9
Andersen, Henning Haahr and Kaneda, Masaharu
Rigidity of tilting modules
Mosc. Math. J. 11 (2011) 1–39
Math Reviews MR2808210
10
Andersen, Henning Haahr and Stroppel, C. and Tubbenhauer, D.
Cellular structures using U_q-tilting modules
Pacific J. Math. 292 (2018) 21–59
11
Andersen, Henning Haahr and Tubbenhauer, D.
Diagram categories for U_q-tilting modules at roots of unity
Transform. Groups 22 (2017) 29–89
Math Reviews MR3620765
12
Arkhipov, Sergey and Bezrukavnikov, Roman
Perverse sheaves on affine flags and Langlands dual group
Israel J. Math. 170 (2009) 135–183
Math Reviews MR2506322
13
Arkhipov, Sergey and Bezrukavnikov, Roman and Ginzburg, Victor
Quantum groups, the loop Grassmannian, and the Springer resolution
J. Amer. Math. Soc. 17 (2004) 595–678
Math Reviews MR2053952
14
15
Be, Alexander and Bernstein, J. and Deligne, P.
Faisceaux pervers
in Analysis and topology on singular spaces, I (Luminy, 1981)
Astérisque 100 (1982) 5–171
Math Reviews MR751966
16
Be, Alexander and Bezrukavnikov, Roman and Mirković, Ivan
Tilting exercises
Mosc. Math. J. 4 (2004) 547–557
Math Reviews MR2119139
17
Be, Alexander and Ginzburg, Victor and Soergel, Wolfgang
Koszul duality patterns in representation theory
J. Amer. Math. Soc. 9 (1996) 473–527
Math Reviews MR1322847
18
Bezrukavnikov, Roman
Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone
Represent. Theory 7 (2003) 1–18
Math Reviews MR1973365
19
Bezrukavnikov, Roman
Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves
Invent. math. 166 (2006) 327–357
Math Reviews MR2249802
20
Bezrukavnikov, Roman and Yun, Zhiwei
On Koszul duality for Kac-Moody groups
Represent. Theory 17 (2013) 1–98
Math Reviews MR3003920
21
Brundan, Jonathan
On the definition of Kac-Moody 2-category
Math. Ann. 364 (2016) 353–372
Math Reviews MR3451390
22
Brundan, Jonathan and Kleshchev, Alexander
Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
Invent. math. 178 (2009) 451–484
Math Reviews MR2551762
23
Buch, Anders S. and Mihalcea, Leonardo C.
Curve neighborhoods of Schubert varieties
J. Differential Geom. 99 (2015) 255–283
Math Reviews MR3302040
24
Chuang, Joseph and Rouquier, Raphaël
Derived equivalences for symmetric groups and sl_2-categorification
Ann. of Math. 167 (2008) 245–298
Math Reviews MR2373155
25
Cline, E. and Parshall, B. and Scott, L.
Finite-dimensional algebras and highest weight categories
J. reine angew. Math. 391 (1988) 85–99
Math Reviews MR961165
26
Collingwood, David H. and Irving, Ronald S.
A decomposition theorem for certain self-dual modules in the category O
Duke Math. J. 58 (1989) 89–102
Math Reviews MR1016415
27
Donkin, Stephen
On tilting modules for algebraic groups
Math. Z. 212 (1993) 39–60
Math Reviews MR1200163
28
29
Elias, Ben and Khovanov, Mikhail
Diagrammatics for Soergel categories
Int. J. Math. Math. Sci. (2010) Art. ID 978635
Math Reviews MR3095655
30
Elias, Ben and Williamson, Geordie
The Hodge theory of Soergel bimodules
Ann. of Math. 180 (2014) 1089–1136
Math Reviews MR3245013
31
Elias, Ben and Williamson, Geordie
Soergel calculus
Represent. Theory 20 (2016) 295–374
Math Reviews MR3555156
32
Erdmann, Karin
Tensor products and dimensions of simple modules for symmetric groups
Manuscripta Math. 88 (1995) 357–386
Math Reviews MR1359703
33
Fiebig, Peter
Lusztig's conjecture as a moment graph problem
Bull. Lond. Math. Soc. 42 (2010) 957–972
Math Reviews MR2740015
34
Fiebig, Peter
An upper bound on the exceptional characteristics for Lusztig's character formula
J. reine angew. Math. 673 (2012) 1–31
Math Reviews MR2999126
35
Finkelberg, Michael and Mirković, Ivan
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 (1999) 81–112
Math Reviews MR1729360
36
Gordon, Robert and Green, Edward L.
Graded Artin algebras
J. Algebra 76 (1982) 111–137
Math Reviews MR659212
37
Görtz, Ulrich
Affine Springer fibers and affine Deligne-Lusztig varieties
in Affine flag manifolds and principal bundles
Trends Math. (2010) 1–50
Math Reviews MR3013026
38
Jantzen, Jens C.
Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne
J. reine angew. Math. 317 (1980) 157–199
Math Reviews MR581341
39
Jantzen, Jens Carsten
Representations of algebraic groups
Amer. Math. Soc., Providence, RI, 2003
Math Reviews MR2015057
40
Jensen, Lars Thorge and Williamson, Geordie
The p-canonical basis for Hecke algebras
in Categorification and higher representation theory
Contemp. Math. 683 (2017) 333–361
Math Reviews MR3611719
41
Juteau, Daniel and Mautner, Carl and Williamson, Geordie
Parity sheaves
J. Amer. Math. Soc. 27 (2014) 1169–1212
Math Reviews MR3230821
42
Kashiwara, Masaki and Schapira, Pierre
Sheaves on manifolds
Springer, Berlin, 1990
Math Reviews MR1074006
43
Kashiwara, Masaki and Schapira, Pierre
Categories and sheaves
Springer, Berlin, 2006
Math Reviews MR2182076
44
Kashiwara, Masaki and Tanisaki, Toshiyuki
Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case
Duke Math. J. 84 (1996) 771–813
Math Reviews MR1408544
45
Kazhdan, D. and Lusztig, George
Tensor structures arising from affine Lie algebras. I
J. Amer. Math. Soc. 6 (1993) 905–947 ; ``Tensor structures arising from affine Lie algebras. II'', J. Amer. Math. Soc. 6 (1993), 949–1011; ``Tensor structures arising from affine Lie algebras. III'', J. Amer. Math. Soc. 7 (1994), 335–381; ``Tensor structures arising from affine Lie algebras. IV'', J. Amer. Math. Soc. 7 (1994), 383–453
Math Reviews MR1239506
46
Kildetoft, Tobias
Decomposition of Tensor Products Involving a Steinberg Module
Algebr. Represent. Theory 20 (2017) 951–975
Math Reviews MR3669165
47
Kumar, Shrawan
Kac-Moody groups, their flag varieties and representation theory
Birkhäuser, 2002
Math Reviews MR1923198
48
Libedinsky, Nicolas
Sur la catégorie des bimodules de Soergel
J. Algebra 320 (2008) 2675–2694
Math Reviews MR2441994
49
50
Lusztig, George
Some problems in the representation theory of finite Chevalley groups
in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979)
Proc. Sympos. Pure Math. 37 (1980) 313–317
Math Reviews MR604598
51
Lusztig, George
Modular representations and quantum groups
in Classical groups and related topics (Beijing, 1987)
Contemp. Math. 82 (1989) 59–77
Math Reviews MR982278
52
Lusztig, George
Monodromic systems on affine flag manifolds
Proc. Roy. Soc. London Ser. A 445 (1994) 231–246 errata: Proc. Roy. Soc. London Ser. A 450 (1995), p. 731–732
53
54
Mackaay, Marco and Stošić, Marko and Vaz, Pedro
A diagrammatic categorification of the q-Schur algebra
Quantum Topol. 4 (2013) 1–75
Math Reviews MR2998837
55
Mackaay, Marco and Thiel, Anne-Laure
A diagrammatic categorification of the affine q-Schur algebra S(n,n) for n3
Pacific J. Math. 278 (2015) 201–233
Math Reviews MR3404672
56
Mackaay, Marco and Thiel, Anne-Laure
Categorifications of the extended affine Hecke algebra and the affine q-Schur algebra S(n,r) for 3 r < n
Quantum Topol. 8 (2017) 113–203
57
58
Mathieu, Olivier
Formules de caractères pour les algèbres de Kac-Moody générales
Astérisque 159-160 (1988)
Math Reviews MR980506
59
Mathieu, Olivier
Construction d'un groupe de Kac-Moody et applications
Compos. math. 69 (1989) 37–60
Math Reviews MR986812
60
Mathieu, Olivier
On the dimension of some modular irreducible representations of the symmetric group
Lett. Math. Phys. 38 (1996) 23–32
Math Reviews MR1401053
61
62
Milne, James S.
Étale cohomology
Princeton Univ. Press, Princeton, N.J., 1980
Math Reviews MR559531
63
Nadler, David
Perverse sheaves on real loop Grassmannians
Invent. math. 159 (2005) 1–73
Math Reviews MR2142332
64
Neeman, Amnon
Triangulated categories
Princeton Univ. Press, Princeton, NJ, 2001
Math Reviews MR1812507
65
Pappas, G. and Rapoport, M.
Twisted loop groups and their affine flag varieties
Adv. Math. 219 (2008) 118–198
Math Reviews MR2435422
66
67
Rousseau, Guy
Groupes de Kac-Moody déployés sur un corps local II. Masures ordonnées
Bull. Soc. Math. France 144 (2016) 613–692
Math Reviews MR3562609
68
Soergel, Wolfgang
Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe
J. Amer. Math. Soc. 3 (1990) 421–445
Math Reviews MR1029692
69
Soergel, Wolfgang
Gradings on representation categories
in Proceedings of the International Congress of Mathematicians, Vol.1, 2 (Zürich, 1994)
(1995) 800–806
Math Reviews MR1403980
70
Soergel, Wolfgang
Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules
Represent. Theory 1 (1997) 83–114
Math Reviews MR1444322
71
Soergel, Wolfgang
Character formulas for tilting modules over Kac-Moody algebras
Represent. Theory 2 (1998) 432–448
Math Reviews MR1663141
72
Soergel, Wolfgang
On the relation between intersection cohomology and representation theory in positive characteristic
J. Pure Appl. Algebra 152 (2000) 311–335
Math Reviews MR1784005
73
Soergel, Wolfgang
Langlands' philosophy and Koszul duality
in Algebra—representation theory (Constanta, 2000)
NATO Sci. Ser. II Math. Phys. Chem. 28 (2001) 379–414
Math Reviews MR1858045
74
Soergel, Wolfgang
Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen
J. Inst. Math. Jussieu 6 (2007) 501–525
Math Reviews MR2329762
75
Springer, T. A.
Quelques applications de la cohomologie d'intersection
Séminaire Bourbaki, vol. 1981/1982, exposé no 589, Astérisque 92 (1982) 249–273
Math Reviews MR689533
76
Tits, Jacques
Groupes associés aux algèbres de Kac-Moody
Séminaire Bourbaki, vol. 1988/1989, exposé no 700, Astérisque 177–178 (1989) 7–31
Math Reviews MR1040566
77
Weidner, J.
Modular equivariant formality
J. Amer. Math. Soc. 30 (2017) 1023–1046
78
Williamson, Geordie
Algebraic representations and constructible sheaves
Jpn. J. Math. 12 (2017) 211–259
79
Williamson, Geordie
Schubert calculus and torsion explosion
J. Amer. Math. Soc. 30 (2017) 1023–1046
80
Williamson, Geordie and Braden, Tom
Modular intersection cohomology complexes on flag varieties
Math. Z. 272 (2012) 697–727
Math Reviews MR2995137