Parity sheaves and tilting modules
[Faisceaux de parité et modules basculants]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 257-275.

Nous montrons que la correspondance de Satake géométrique fait correspondre les modules basculants aux faisceaux à parité sur la grassmannienne affine, lorsque la caractéristique est plus grande qu'une borne explicite.

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

Publié le :
DOI : 10.24033/asens.2282
Classification : 20G05, 22E57, 32S60
Keywords: Perverse sheaves, geometric Satake equivalence, parity sheaves, tilting modules.
Mot clés : Faisceaux pervers, équivalence de Satake géométrique, faisceaux de parité, modules basculants. 
@article{ASENS_2016__49_2_257_0,
     author = {Juteau, Daniel and Mautner, Carl and Williamson, Geordie},
     title = {Parity sheaves and tilting modules},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {257--275},
     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.2282},
     mrnumber = {3481350},
     zbl = {1402.14023},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2282/}
}
TY  - JOUR
AU  - Juteau, Daniel
AU  - Mautner, Carl
AU  - Williamson, Geordie
TI  - Parity sheaves and tilting modules
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 257
EP  - 275
VL  - 49
IS  - 2
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2282/
DO  - 10.24033/asens.2282
LA  - en
ID  - ASENS_2016__49_2_257_0
ER  - 
%0 Journal Article
%A Juteau, Daniel
%A Mautner, Carl
%A Williamson, Geordie
%T Parity sheaves and tilting modules
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 257-275
%V 49
%N 2
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2282/
%R 10.24033/asens.2282
%G en
%F ASENS_2016__49_2_257_0
Juteau, Daniel; Mautner, Carl; Williamson, Geordie. Parity sheaves and tilting modules. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 257-275. doi : 10.24033/asens.2282. http://www.numdam.org/articles/10.24033/asens.2282/

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

Beilinson, A.; Bezrukavnikov, R.; Mirković, I. Tilting exercises, Mosc. Math. J., Volume 4 (2004), p. 547-557, 782 (ISSN: 1609-3321) | DOI | MR | Zbl

Benson, D. J.; Carlson, J. F. Nilpotent elements in the Green ring, J. Algebra, Volume 104 (1986), pp. 329-350 (ISSN: 0021-8693) | DOI | MR | Zbl

Beĭlinson, A.; Drinfel'd, V. Quantization of Hitchin integrable system and Hecke eigensheaves (2000) (preprint http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf )

Białynicki-Birula, A. Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 24 (1976), pp. 667-674 (ISSN: 0001-4117) | MR | Zbl

Bourbaki, N., Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968 ( ; réédition Springer, 2002) | MR | Zbl

Braden, T. Hyperbolic localization of intersection cohomology, Transform. Groups, Volume 8 (2003), pp. 209-216 (ISSN: 1083-4362) | DOI | MR | Zbl

Brylinski, R. K. Limits of weight spaces, Lusztig's q-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc., Volume 2 (1989), pp. 517-533 (ISSN: 0894-0347) | DOI | MR | Zbl

Donkin, S., Lecture Notes in Math., 1140, Springer, Berlin, 1985, 254 pages (ISBN: 3-540-15668-2) | MR | Zbl

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

Donkin, S., London Math. Soc. Lecture Note Series, 253, Cambridge Univ. Press, Cambridge, 1998, 179 pages (ISBN: 0-521-64558-1) | DOI | MR | Zbl

Gaussent, S.; Littelmann, P. LS galleries, the path model, and MV cycles, Duke Math. J., Volume 127 (2005), pp. 35-88 (ISSN: 0012-7094) | DOI | MR | Zbl

Hesselink, W. H., Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980) (Lecture Notes in Math.), Volume 867, Springer, Berlin, 1981, pp. 55-89 | DOI | MR | Zbl

Jantzen, J. C., Mathematical Surveys and Monographs, 107, Amer. Math. Soc., Providence, RI, 2003, 576 pages (ISBN: 0-8218-3527-0) | MR | Zbl

Jantzen, J. C., Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) (Progr. Math.), Volume 95, Birkhäuser, 1991, pp. 289-315 | 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. Modular representations of reductive groups and geometry of affine Grassmannians (preprint arXiv:0804.2041 )

Kaneda, M. Based modules and good filtrations in algebraic groups, Hiroshima Math. J., Volume 28 (1998), pp. 337-344 http://projecteuclid.org/euclid.hmj/1206126765 (ISSN: 0018-2079) | DOI | MR | Zbl

Knop, F.; Kraft, H.; Luna, D.; Vust, T., Algebraische Transformationsgruppen und Invariantentheorie (DMV Sem.), Volume 13, Birkhäuser, 1989, pp. 63-75 | DOI | MR | Zbl

Kumar, S., Progress in Math., 204, Birkhäuser, 2002, 606 pages (ISBN: 0-8176-4227-7) | DOI | MR | Zbl

Littelmann, P. Good filtrations and decomposition rules for representations with standard monomial theory, J. reine angew. Math., Volume 433 (1992), pp. 161-180 (ISSN: 0075-4102) | DOI | MR | Zbl

Lübeck, F. Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., Volume 4 (2001), pp. 135-169 (ISSN: 1461-1570) | DOI | MR | Zbl

Lusztig, G., Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101, Soc. Math. France, Paris, 1983, pp. 208-229 | Numdam | MR | Zbl

Mathieu, O. Filtrations of G-modules, Ann. Sci. École Norm. Sup., Volume 23 (1990), pp. 625-644 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

McNinch, G. J. Semisimplicity of exterior powers of semisimple representations of groups, J. Algebra, Volume 225 (2000), pp. 646-666 (ISSN: 0021-8693) | DOI | MR | Zbl

Malkin, A.; Ostrik, V.; Vybornov, M. The minimal degeneration singularities in the affine Grassmannians, Duke Math. J., Volume 126 (2005), pp. 233-249 (ISSN: 0012-7094) | DOI | MR | Zbl

Mirković, I.; Vilonen, K. Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math., Volume 166 (2007), pp. 95-143 (ISSN: 0003-486X) | DOI | MR | Zbl

Paradowski, J., Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991) (Proc. Sympos. Pure Math.), Volume 56, Amer. Math. Soc., Providence, RI, 1994, pp. 93-108 | MR | Zbl

Polo, P. Modules associés aux variétés de Schubert, Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi (1993), pp. 155-171 | 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

Sumihiro, H. Equivariant completion, J. Math. Kyoto Univ., Volume 14 (1974), pp. 1-28 (ISSN: 0023-608X) | MR | Zbl

Wang, J. P. Sheaf cohomology on G/B and tensor products of Weyl modules, J. Algebra, Volume 77 (1982), pp. 162-185 (ISSN: 0021-8693) | DOI | MR | Zbl

Cité par Sources :