Géométrie des variétés de Deligne-Lusztig, décompositions, cohomologie modulo \ell et représentations modulaires

Abstract : This work is a contribution to the modular representation theory of finite reductive groups. As in the ordinary setting, we are mainly interested in geometric constructions of the representations by means of the cohomology of Deligne-Lusztig varieties. We start by studying a Deodhar-type decomposition that we use to locate a certain class of representations, the so-called Gelfand-Graev modules and some of their generalizations. More precise results are obtained for varieties associated to some short-length regular elements. The case of Coxeter elements holds an important place in this work: for these specific elements we give an explicit construction of a complex representing the cohomology of the corresponding varieties, leading to a proof of the geometric version of Broué's conjecture for some prime numbers. We also deduce the Brauer tree of the principal block in this case, which settles a conjecture of Hiss, Lübeck and Malle. Both of these results rely on the assumption that the cohomology is torsion-free, which is shown to hold for several classical and exceptional groups.
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Mathematics [math]. Université de Franche-Comté, 2010. French


https://tel.archives-ouvertes.fr/tel-00492848
Contributor : Olivier Dudas <>
Submitted on : Thursday, June 17, 2010 - 11:41:06 AM
Last modification on : Thursday, June 17, 2010 - 12:44:59 PM

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Olivier Dudas. Géométrie des variétés de Deligne-Lusztig, décompositions, cohomologie modulo \ell et représentations modulaires. Mathematics [math]. Université de Franche-Comté, 2010. French. <tel-00492848>

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