An Iwahori-Whittaker model for the Satake category

Bezrukavnikov, Roman; Gaitsgory, Dennis; Mirković, Ivan; Riche, Simon; Rider, Laura

doi:10.5802/jep.104

An Iwahori-Whittaker model for the Satake category
[Un modèle d’Iwahori-Whittaker pour la catégorie de Satake]

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

Journal de l’École polytechnique - Mathématiques, Tome 6 (2019), pp. 707-735.

Résumé
Abstract

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.

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.

Reçu le : 2018-07-09
Accepté le : 2019-07-11
Publié le : 2019-09-25

Zbl | 1 citation dans Numdam

DOI : 10.5802/jep.104

Classification : 20G05
Keywords: Affine Grassmannian, perverse sheaves, geometric Satake equivalence, tilting modules, parity sheaves
Mot clés : Grassmannienne affine, faisceaux pervers, équivalence de Satake géométrique, modules basculants, faisceaux à parité

Affiliations des auteurs :

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

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     title = {An {Iwahori-Whittaker} model for the {Satake} category},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique - Math\'ematiques},
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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://www.numdam.org/articles/10.5802/jep.104/

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