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Geometric braid group action on derived categories of coherent sheaves. With a joint appendix with Roman Bezrukavnikov. (English) Zbl 1156.14014
Summary: We give, for semi-simple groups without factors of type $$\mathbf{G}_2$$, a geometric construction of a braid group action on $$\mathcal{D}^b \text{Coh}(\widetilde{\mathfrak{g}})$$ extending the action constructed by Bezrukavnikov, Mirković and Rumynin in the context of localization in positive characteristic. It follows that this action extends to characteristic zero, where it also has some nice representation-theoretic interpretations. The argument uses a presentation of the affine braid group analogous to the “Bernstein presentation” of the corresponding Hecke algebra (this presentation was suggested by Lusztig; it is worked out in the appendix, written jointly with Roman Bezrukavnikov).

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14M15 Grassmannians, Schubert varieties, flag manifolds 20F55 Reflection and Coxeter groups (group-theoretic aspects) 18E30 Derived categories, triangulated categories (MSC2010)
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