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Geometric braid group action on derived categories of coherent sheaves


Author: Simon Riche; with a joint appendix with Roman Bezrukavnikov
Journal: Represent. Theory 12 (2008), 131-169
MSC (2000): Primary 14M15; Secondary 20F55, 18E30
DOI: https://doi.org/10.1090/S1088-4165-08-00325-7
Published electronically: March 10, 2008
MathSciNet review: 2390670
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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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Additional Information

Simon Riche
Affiliation: Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu (UMR 7586 du CNRS), Équipe d’Analyse Algébrique, 175, rue du Chevaleret, 75013 Paris, France
Email: riche@math.jussieu.fr

Roman Bezrukavnikov
Affiliation: Massachusetts Institute of Technology, Cambridge, Massachusetts
Email: bezrukav@math.mit.edu

DOI: https://doi.org/10.1090/S1088-4165-08-00325-7
Received by editor(s): March 12, 2007
Received by editor(s) in revised form: July 23, 2007
Published electronically: March 10, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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