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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Coxeter orbits and Brauer trees III


Authors: Olivier Dudas and Raphaël Rouquier
Journal: J. Amer. Math. Soc. 27 (2014), 1117-1145
MSC (2010): Primary 20C33; Secondary 18E30, 20C20
Published electronically: March 25, 2014
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Abstract: This article is the final one of a series of articles on certain blocks of modular representations of finite groups of Lie type and the associated geometry. We prove the conjecture of Broué on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements. We also prove a conjecture of Hiß, Lübeck, and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer tree (in particular, the decomposition matrix) of the principal $ \ell $-block of $ E_7(q)$ when $ \ell \mid \Phi _{18}(q)$ and $ E_8(q)$ when $ \ell \mid \Phi _{18}(q)$ or $ \ell \mid \Phi _{30}(q)$.


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Additional Information

Olivier Dudas
Affiliation: Université Denis Diderot – Paris 7, UFR de Mathématiques, Institut de Mathématiques de Jussieu, Case 7012, 75205 Paris Cedex 13, France
Email: dudas@math.jussieu.fr

Raphaël Rouquier
Affiliation: Department of Mathematics, UCLA, Box 951555, Los Angeles, California 90095-1555
Email: rouquier@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2014-00791-8
PII: S 0894-0347(2014)00791-8
Keywords: Finite reductive groups, Brauer trees, Deligne-Lusztig varieties, Derived categories, Brou\'e's conjecture
Received by editor(s): October 16, 2012
Received by editor(s) in revised form: August 16, 2013
Published electronically: March 25, 2014
Additional Notes: The first author was supported by the EPSRC, Project No EP/H026568/1 and by Magdalen College, Oxford.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.