Decomposition numbers for perverse sheaves

Juteau, Daniel

Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1177-1229.

Le but de cet article est de poser les fondations pour les nombres de décomposition des faisceaux pervers, de donner quelques méthodes pour les calculer dans des cas simples et de les déterminer explicitement dans deux situations : pour une singularité simple (kleinienne) de surface et pour l’adhérence de l’orbite nilpotente non-triviale minimale dans une algèbre de Lie simple.

Ce travail a des applications dans la théorie des représentations modulaires, pour les groupes de Weyl en utilisant le cône nilpotent de l’algèbre de Lie semi-simple correspondante, et pour les schémas en groupes réductifs en utilisant la grassmannienne affine du dual de Langlands.

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial nilpotent orbit in a simple Lie algebra.

This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.

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