Group theory/Lie algebras
A non-perverse Soergel bimodule in type A
Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 853-858.

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.

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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Published online:
DOI: 10.1016/j.crma.2017.07.011
Libedinsky, Nicolas 1, 2; Williamson, Geordie 2

1 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Las Palmeras 3425, Nuñoa, Santiago, Chile
2 University of Sydney, Sydney, NSW, 2006, Australia
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Libedinsky, Nicolas; Williamson, Geordie. A non-perverse Soergel bimodule in type A. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 853-858. doi : 10.1016/j.crma.2017.07.011. http://www.numdam.org/articles/10.1016/j.crma.2017.07.011/

[1] Achar, P.N.; Makisumi, S.; Riche, S.; Williamson, G. Koszul duality for Kac–Moody groups and characters of tilting modules, 2017 (preprint) | arXiv

[2] Achar, P.; Riche, S. Modular perverse sheaves on flag varieties III: positivity conditions (preprint) | arXiv

[3] Elias, B. Thicker Soergel calculus in type A, Proc. Lond. Math. Soc. (3), Volume 112 (2016) no. 5, pp. 924-978

[4] Elias, B.; Williamson, G. The Hodge theory of Soergel bimodules, Ann. of Math. (2), Volume 180 (2014) no. 3, pp. 1089-1136

[5] Elias, B.; Williamson, G. Soergel calculus, Represent. Theory, Volume 20 (2016), pp. 295-374

[6] He, X.; Williamson, G. Soergel calculus and Schubert calculus (preprint) | arXiv

[7] Juteau, D.; Mautner, C.; Williamson, G. Parity sheaves, J. Amer. Math. Soc., Volume 27 (2014) no. 4, pp. 1169-1212

[8] Juteau, D.; Mautner, C.; Williamson, G. Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016) no. 2, pp. 257-275

[9] Jensen, T.; Williamson, G. The p-canonical basis for Hecke algebras, Categorification in Geometry, Topology and Physics, Contemp. Math., 2017, pp. 123-161

[10] Lusztig, G.; Williamson, G. Billiards and tilting characters for SL3, 2018 (preprint) | arXiv

[11] Maunter, C.; Riche, S. Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture, J. Eur. Math. Soc. (2015) (preprint in press) | arXiv

[12] Riche, S.; Williamson, G. Tilting modules and the p-canonical basis, Astérisque (2017) (preprint in press) | arXiv

[13] Soergel, W. Kazhdan–Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory, Volume 1 (1997), pp. 83-114 (electronic)

[14] Williamson, G. Singular Soergel bimodules, Int. Math. Res. Not. (2011) no. 20, pp. 4555-4632

[15] Williamson, G. Algebraic representations and constructible sheaves, Jpn. J. Math. (2017) (Takagi lecture in press) | arXiv

[16] Williamson, G. Schubert calculus and torsion explosion, J. Amer. Math. Soc., Volume 30 (2017) no. 4, pp. 1023-1046

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