Block decomposition via the geometric Satake equivalence
[Décomposition en blocs via l’équivalence de Satake géométrique]
Zabeth, Emilien1
1Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand (France)
Annales de l'Institut Fourier, Online first, 73 p.

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group $\mathbf{G}$ over a field of positive characteristic $\ell $ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith–Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an $\ell ^{\text{th}}$ root of unity.

Nous donnons une nouvelle preuve pour la description des blocs de la catégorie des représentations d’un groupe algébrique réductif sur un corps de caractéristique positive $\ell $ (originellement due à Donkin), en travaillant dans la catégorie de Satake du groupe dual de Langlands et en appliquant la théorie de Smith–Treumann telle que développée par Riche et Williamson. Du côté de la théorie des représentations, nos méthodes nous permettent de donner une borne pour la longueur minimale d’une chaîne reliant deux poids dans le même bloc, et de donner une nouvelle preuve de la décomposition en blocs d’un groupe quantique à une racine $\ell ^{\text{ème}}$ de l’unité.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3755
Classification : 20G05, 14F08, 22E57
Keywords: Kazhdan–Lusztig polynomials, reductive algebraic groups, perverse sheaves
Mots-clés : polynômes de Kazhdan–Lusztig, groupes algébriques réductifs, faisceaux pervers

Zabeth, Emilien  1

1 Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand (France) 
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Zabeth, Emilien. Block decomposition via the geometric Satake equivalence. Annales de l'Institut Fourier, Online first, 73 p.

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