An Iwahori-Whittaker model for the Satake category

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

doi:10.5802/jep.104

Roman Bezrukavnikov ¹; Dennis Gaitsgory ²; Ivan Mirković ³; Simon Riche ⁴; Laura Rider 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, Volume 6 (2019), pp. 707-735.

Abstract
Résumé

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.

Received: 2018-07-08
Accepted: 2019-07-10
Published online: 2019-09-24

Zbl

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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{JEP_2019__6__707_0,
     author = {Roman Bezrukavnikov and Dennis Gaitsgory and Ivan Mirkovi\'c and Simon Riche and Laura Rider},
     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},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.104},
     zbl = {07114040},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.104/}
}

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Roman Bezrukavnikov; Dennis Gaitsgory; Ivan Mirković; Simon Riche; Laura Rider. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.104/

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