[Nombres de décomposition pour les faisceaux pervers]
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.
Keywords: Perverse sheaves, intersection cohomology, integral cohomology, t-structures, torsion theories, decomposition matrices, simple singularities, minimal nilpotent orbits
Mot clés : faisceaux pervers, cohomologie d’intersection, cohomologie entière, t-structures, théories de torsion, matrices de décomposition, singularités simples, orbites nilpotentes minimales
@article{AIF_2009__59_3_1177_0,
author = {Juteau, Daniel},
title = {Decomposition numbers for perverse sheaves},
journal = {Annales de l'Institut Fourier},
pages = {1177--1229},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {3},
year = {2009},
doi = {10.5802/aif.2461},
zbl = {1187.14022},
mrnumber = {2543666},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2461/}
}
TY - JOUR AU - Juteau, Daniel TI - Decomposition numbers for perverse sheaves JO - Annales de l'Institut Fourier PY - 2009 SP - 1177 EP - 1229 VL - 59 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2461/ DO - 10.5802/aif.2461 LA - en ID - AIF_2009__59_3_1177_0 ER -
Juteau, Daniel. Decomposition numbers for perverse sheaves. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1177-1229. doi : 10.5802/aif.2461. http://www.numdam.org/articles/10.5802/aif.2461/
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