Modular perverse sheaves  on flag varieties I:  tilting and parity sheaves

Achar, Pramod N.; Riche, Simon

doi:10.24033/asens.2284

Modular perverse sheaves on flag varieties I: tilting and parity sheaves
[Faisceaux pervers modulaires sur les variétés de drapeaux I : faisceaux basculants et faisceaux à parité]

Achar, Pramod N. ; Riche, Simon

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 325-370.

Résumé
Abstract

Dans cet article nous démontrons que la catégorie des complexes à parité sur la variété de drapeaux d'un groupe réductif complexe connexe $G$ est une « version graduée » de la catégorie des faisceaux pervers basculants sur la variété de drapeaux du groupe dual $\overset{ˇ}{G}$ , pour tout corps de coefficients dont la caractéristique est bonne pour $G$ . Nous en déduisons des conséquences sur la catégorie $𝒪$ modulaire de Soergel, et sur le calcul des multiplicités et des nombres de décomposition dans la catégorie des faisceaux pervers.

In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group $G$ is a “graded version” of the category of tilting perverse sheaves on the flag variety of the dual group $\overset{ˇ}{G}$ , for any field of coefficients whose characteristic is good for $G$ . We derive some consequences on Soergel's modular category $𝒪$ , and on multiplicities and decomposition numbers in the category of perverse sheaves.

MR Zbl

DOI : 10.24033/asens.2284

Classification : 14M15; 14F05, 20G40.
Keywords: Flag varieties, reductive algebraic groups, perverse sheaves, parity sheaves, Soergel bimodules, quasi-hereditary categories, modular reduction.
Mot clés : Variétés de drapeaux, groupes algébriques réductifs, faisceaux pervers, faisceaux à parité, bimodules de Soergel, catégories quasi-héréditaires, réduction modulaire.

@article{ASENS_2016__49_2_325_0,
     author = {Achar, Pramod N. and Riche, Simon},
     title = {Modular perverse sheaves  on flag varieties {I:}  tilting and parity sheaves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {325--370},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {2},
     year = {2016},
     doi = {10.24033/asens.2284},
     mrnumber = {3481352},
     zbl = {1386.14178},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2284/}
}

TY  - JOUR
AU  - Achar, Pramod N.
AU  - Riche, Simon
TI  - Modular perverse sheaves  on flag varieties I:  tilting and parity sheaves
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 325
EP  - 370
VL  - 49
IS  - 2
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2284/
DO  - 10.24033/asens.2284
LA  - en
ID  - ASENS_2016__49_2_325_0
ER  -

%0 Journal Article
%A Achar, Pramod N.
%A Riche, Simon
%T Modular perverse sheaves  on flag varieties I:  tilting and parity sheaves
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 325-370
%V 49
%N 2
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2284/
%R 10.24033/asens.2284
%G en
%F ASENS_2016__49_2_325_0

Achar, Pramod N.; Riche, Simon. Modular perverse sheaves  on flag varieties I:  tilting and parity sheaves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 325-370. doi : 10.24033/asens.2284. http://www.numdam.org/articles/10.24033/asens.2284/

Bibliographie
Cité par

Achar, P. N.; Henderson, A.; Juteau, D.; Riche, S. Weyl group actions on the Springer sheaf, Proc. Lond. Math. Soc., Volume 108 (2014), pp. 1501-1528 (ISSN: 0024-6115) | DOI | MR | Zbl

Achar, P. N.; Rider, L. Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture (preprint arXiv:1305.1684, to appear in Acta Math ) | MR

Achar, P. N.; Riche, S. Modular perverse sheaves on flag varieties II: Koszul duality and formality (preprint arXiv:1401.7256, to appear in Duke Math. J ) | MR

Achar, P. N.; Riche, S. Modular perverse sheaves on flag varieties III: positivity conditions (preprint arXiv:1408.4189 ) | MR

Beĭlinson, A.; Bernstein, J., Representation theory of reductive groups (Park City, Utah, 1982) (Progr. Math.), Volume 40, Birkhäuser, 1983, pp. 35-52 | DOI | MR | Zbl

Beĭlinson, A.; Bernstein, J.; Deligne, P., Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Soc. Math. France, Paris, 1982, pp. 5-171 | Numdam | MR | Zbl

Beĭlinson, A.; Bezrukavnikov, R.; Mirković, I. Tilting exercises, Mosc. Math. J., Volume 4 (2004), pp. 547-557 (ISSN: 1609-3321) | DOI | MR | Zbl

Bezrukavnikov, R.; Braverman, A.; Mirković, I. Some results about geometric Whittaker model, Adv. Math., Volume 186 (2004), pp. 143-152 (ISSN: 0001-8708) | DOI | MR | Zbl

Bernstein, J., Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) (Progr. Math.), Volume 92, Birkhäuser, 1990, pp. 417-423 | MR | Zbl

Beĭlinson, A., $K$ -theory, arithmetic and geometry (Moscow, 1984–1986) (Lecture Notes in Math.), Volume 1289, Springer, Berlin, 1987, pp. 27-41 | DOI | MR | Zbl

Beĭlinson, A.; Ginzburg, V.; Soergel, W. Koszul duality patterns in representation theory, J. Amer. Math. Soc., Volume 9 (1996), pp. 473-527 (ISSN: 0894-0347) | DOI | MR | Zbl

Bernstein, J.; Lunts, V., Lecture Notes in Math., 1578, Springer, Berlin, 1994, 139 pages (ISBN: 3-540-58071-9) | MR | Zbl

Bardsley, P.; Richardson, R. W. Étale slices for algebraic transformation groups in characteristic $p$ , Proc. London Math. Soc., Volume 51 (1985), pp. 295-317 (ISSN: 0024-6115) | DOI | MR | Zbl

Bezrukavnikov, R.; Yun, Z. On Koszul duality for Kac-Moody groups, Represent. Theory, Volume 17 (2013), pp. 1-98 (ISSN: 1088-4165) | DOI | MR | Zbl

Demazure, M. Invariants symétriques entiers des groupes de Weyl et torsion, Invent. math., Volume 21 (1973), pp. 287-301 (ISSN: 0020-9910) | DOI | MR | Zbl

Donkin, S. On tilting modules for algebraic groups, Math. Z., Volume 212 (1993), pp. 39-60 (ISSN: 0025-5874) | DOI | MR | Zbl

Elias, B.; Williamson, G. Soergel calculus (preprint arXiv:1309.0865 ) | MR

Gordon, R.; Green, E. L. Graded Artin algebras, J. Algebra, Volume 76 (1982), pp. 111-137 (ISSN: 0021-8693) | DOI | MR | Zbl

Juteau, D.; Mautner, C.; Williamson, G. Parity sheaves, J. Amer. Math. Soc., Volume 27 (2014), pp. 1169-1212 (ISSN: 0894-0347) | DOI | MR | Zbl

Juteau, D. Decomposition numbers for perverse sheaves, Ann. Inst. Fourier (Grenoble), Volume 59 (2009), pp. 1177-1229 http://aif.cedram.org/item?id=AIF_2009__59_3_1177_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Jensen, L. T.; Williamson, G. The $p$ -canonical basis for Hecke algebras (preprint arXiv:1510.01556 ) | MR

Kazhdan, D.; Lusztig, G., Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) (Proc. Sympos. Pure Math., XXXVI), Amer. Math. Soc., Providence, R.I., 1980, pp. 185-203 | MR | Zbl

Mathieu, O., Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997) (Adv. Stud. Pure Math.), Volume 26, Math. Soc. Japan, Tokyo, 2000, pp. 145-212 | DOI | MR | Zbl

Ringel, C. M. The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z., Volume 208 (1991), pp. 209-223 (ISSN: 0025-5874) | DOI | MR | Zbl

Riche, S.; Soergel, W.; Williamson, G. Modular Koszul duality, Compos. Math., Volume 150 (2014), pp. 273-332 (ISSN: 0010-437X) | DOI | MR | Zbl

Soergel, W. On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra, Volume 152 (2000), pp. 311-335 (ISSN: 0022-4049) | DOI | MR | Zbl

Soergel, W. $𝔫$ -cohomology of simple highest weight modules on walls and purity, Invent. math., Volume 98 (1989), pp. 565-580 (ISSN: 0020-9910) | DOI | MR | Zbl

Soergel, W. Kategorie $𝒪$ , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc., Volume 3 (1990), pp. 421-445 (ISSN: 0894-0347) | DOI | MR | Zbl

Springer, T. A., Bourbaki Seminar, Vol. 1981/1982 (Astérisque), Volume 92, Soc. Math. France, Paris, 1982, pp. 249-273 | Numdam | MR | Zbl

Springer, T. A.; Steinberg, R., Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) (Lecture Notes in Math.), Volume 131, Springer, Berlin, 1970, pp. 167-266 | DOI | MR | Zbl

Verdier, J.-L., Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101, Soc. Math. France, Paris, 1983, pp. 332-364 | Numdam | MR | Zbl

Weidner, J. Modular equivariant formality ( http://www.freidok.uni-freiburg.de/volltexte/9637/ ) | Zbl

Weidner, J. Grassmannians and Koszul duality, Math. Z., Volume 278 (2014), pp. 1033-1064 (ISSN: 0025-5874) | DOI | MR | Zbl

Williamson, G. A reducible characteristic variety in type $A$ (preprint arXiv:1405.3479 ) | MR

Williamson, G. Schubert calculus and torsion explosion (with an appendix joint with A. Kontorovich and P. J. McNamara, preprint arXiv:1309.5055 ) | MR

Yun, Z. Weights of mixed tilting sheaves and geometric Ringel duality, Selecta Math. (N.S.), Volume 14 (2009), pp. 299-320 (ISSN: 1022-1824) | DOI | MR | Zbl

Cité par Sources :