Geometric braid group action on derived categories of coherent sheaves

International audienceIn this paper we give, for semi-simple groups without factors of type G_2, a geometric construction of a braid group action on the derived category of coherent sheaves on the Grothendieck resolution extending the action constructed by Bezrukavnikov, Mirkovic and Rumynin. 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, joint with Roman Bezrukavnikov)