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Comptes Rendus Mathématique
Volume 355, n° 8
pages 853-858 (août 2017)
Doi : 10.1016/j.crma.2017.07.011
Received : 21 July 2017 ;  accepted : 25 July 2017
A non-perverse Soergel bimodule in type A
Un bimodule de Soergel non pervers de type A
 

Nicolas Libedinsky a, b , Geordie Williamson b
a Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Las Palmeras 3425, Nuñoa, Santiago, Chile 
b University of Sydney, Sydney, NSW, 2006, Australia 

Abstract

A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan–Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A .

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Résumé

L'étude de l'anneau des endomorphismes des bimodules de Soergel indécomposables est une question importante. En caractéristique zéro, tous les endomorphismes de degré zero sont des isomorphismes (comme démontré par Elias et le deuxième auteur). Ceci implique les conjectures de Kazhdan–Lusztig. Plus récemment, en caractéristique positive, de nombreux exemples ont été trouvés d'endomorphismes de degré zero qui ne sont pas des isomorphismes. Ceci donne des contre-exemples aux bornes dans la conjecture de Lusztig. Dans cette Note, nous prouvons l'existence de bimodules de Soergel indécomposables, de type A , ayant un endomorphisme de degré négatif. Ceci prouve l'existence d'un faisceau de parité non pervers de type A .

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