Geometric braid group action on derived categories of coherent sheaves

Riche, Simon

doi:10.1090/S1088-4165-08-00325-7

Geometric braid group action on derived categories of coherent sheaves
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by Simon Riche; \\ with a joint appendix with Roman Bezrukavnikov

Represent. Theory 12 (2008), 131-169

DOI: https://doi.org/10.1090/S1088-4165-08-00325-7

Published electronically: March 10, 2008

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Abstract:

In this paper we give, for semi-simple groups without factors of type $\mathbf {G}_2$, a geometric construction of a braid group action on $\mathcal {D}^b \operatorname {Coh}(\widetilde {\mathfrak {g}})$ 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).

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