Geodesic
An Iwahori-Whittaker model for the Satake category
[Un modèle d’Iwahori-Whittaker pour la catégorie de Satake]
Bezrukavnikov, Roman1 ; Gaitsgory, Dennis2 ; Mirković, Ivan3 ; Riche, Simon4 ; Rider, Laura5
1 Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA
2 Harvard University 1 Oxford St, Cambridge, MA 02138, USA
3 University of Massachusetts Amherst, MA, USA.
4 Université Clermont Auvergne, CNRS, LMBP F-63000 Clermont-Ferrand, France
5 Department of Mathematics, University of Georgia Athens Georgia 30602, USA
Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 707-735.

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In this paper we prove, for G a connected reductive algebraic group satisfying a mild technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of ℚ ℓ , or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As applications, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category, and give a new proof of the property that a tensor product of tilting modules is tilting.

Dans cet article nous montrons, pour G un groupe algébrique réductif connexe satisfaisant à une hypothèse technique mineure, que la catégorie de Satake de G (avec coefficients dans un corps fini, une extension finie des nombres p-adiques, ou l’anneau des entiers d’un tel corps) peut se décrire en termes de faisceaux pervers d’Iwahori-Whittaker sur la grassmannienne affine. Nous en déduisons la démonstration d’une conjecture de Juteau-Mautner-Williamson décrivant les objets basculants dans la catégorie de Satake, et également une nouvelle preuve du fait qu’un produit tensoriel de représentations basculantes est basculant.

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DOI : 10.5802/jep.104
Classification : 20G05
Keywords: Affine Grassmannian, perverse sheaves, geometric Satake equivalence, tilting modules, parity sheaves
Mots-clés : Grassmannienne affine, faisceaux pervers, équivalence de Satake géométrique, modules basculants, faisceaux à parité

Bezrukavnikov, Roman 1 ; Gaitsgory, Dennis 2 ; Mirković, Ivan 3 ; Riche, Simon 4 ; Rider, Laura 5

1 Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA
2 Harvard University 1 Oxford St, Cambridge, MA 02138, USA
3 University of Massachusetts Amherst, MA, USA.
4 Université Clermont Auvergne, CNRS, LMBP F-63000 Clermont-Ferrand, France
5 Department of Mathematics, University of Georgia Athens Georgia 30602, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Bezrukavnikov, Roman and Gaitsgory, Dennis and Mirkovi\'c, Ivan and Riche, Simon and Rider, Laura},
     title = {An {Iwahori-Whittaker} model for the {Satake} category},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {707--735},
     publisher = {\'Ecole polytechnique},
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Bezrukavnikov, Roman; Gaitsgory, Dennis; Mirković, Ivan; Riche, Simon; Rider, Laura. An Iwahori-Whittaker model for the Satake category. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 707-735. doi : 10.5802/jep.104. http://geodesic.mathdoc.fr/articles/10.5802/jep.104/

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