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Affine braid group actions on derived categories of Springer resolutions

Roman Bezrukavnikov;  Simon Riche — 2012

Annales scientifiques de l'École Normale Supérieure

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...

Koszul duality and semisimplicity of Frobenius

Pramod N. Achar;  Simon Riche — 2013

Annales de l’institut Fourier

A fundamental result of Beĭlinson–Ginzburg–Soergel states that on flag varieties and related spaces, a certain modified version of the category of ℓ -adic perverse sheaves exhibits a phenomenon known as . The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue....

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