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Annales scientifiques de l'ENS - Parutions - série 4, 45 (2012)

Parutions < série 4, 45

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 45, fascicule 4 (2012)

Roman Bezrukavnikov, Simon Riche
Affine braid group actions on derived categories of Springer resolutions
Annales scientifiques de l'ENS 45, fascicule 4 (2012), 535-599

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Résumé :
Actions du groupe de tresses affine sur les catégories dérivées des résolutions de Springer
Dans cet article nous construisons et étudions une action du groupe de tresses affine associé à un groupe algébrique semi-simple sur les catégories dérivées de faisceaux cohérents sur diverses variétés liées à la résolution de Springer du cône nilpotent. En particulier, nous décrivons explicitement l'action du groupe de tresses d'Artin. Cette action est une version catégorique de la construction géométrique de l'algèbre de Hecke affine due à Kazhdan-Lusztig et Ginzburg, et est utilisée par le premier auteur et I. Mirković au cours de la preuve des conjectures de Lusztig sur la K-théorie équivariante des fibres de Springer.

Mots-clefs : Groupe de tresses, groupe algébrique réductif, algèbre de Lie, résolution de Springer, algèbre de Hecke affine, dg-schéma, transformée de Fourier-Mukai.

Abstract:
In this paper we construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a ``categorical version'' of Kazhdan-Lusztig-Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.

Keywords: Braid group, reductive algebraic group, Lie algebra, Springer resolution, affine Hecke algebra, dg-scheme, Fourier-Mukai transform.

Class. math. : 20G99; 14M15, 17B20, 18E30, 20F36


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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