An Iwahori-Whittaker model for the Satake category
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 707-735.

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 , 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.

Received:
Accepted:
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DOI: 10.5802/jep.104
Classification: 20G05
Keywords: Affine Grassmannian, perverse sheaves, geometric Satake equivalence, tilting modules, parity sheaves
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
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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, Volume 6 (2019), pp. 707-735. doi : 10.5802/jep.104. http://www.numdam.org/articles/10.5802/jep.104/

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