Affine braid group actions on derived categories of Springer resolutions
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 4, p. 535-599

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.

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.

DOI : https://doi.org/10.24033/asens.2173
Classification:  20G99,  14M15,  17B20,  18E30,  20F36
Keywords: braid group, reductive algebraic group, Lie algebra, Springer resolution, affine Hecke algebra, dg-scheme, Fourier-Mukai transform
@article{ASENS_2012_4_45_4_535_0,
     author = {Bezrukavnikov, Roman and Riche, Simon},
     title = {Affine braid group actions on derived categories of Springer resolutions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {4},
     year = {2012},
     pages = {535-599},
     doi = {10.24033/asens.2173},
     mrnumber = {3059241},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_4_535_0}
}
Bezrukavnikov, Roman; Riche, Simon. Affine braid group actions on derived categories of Springer resolutions. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 4, pp. 535-599. doi : 10.24033/asens.2173. http://www.numdam.org/item/ASENS_2012_4_45_4_535_0/

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