Cohomologie des vari\'{e}t\'{e}s de Deligne-Lusztig
Abstract
We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Brou\'{e}, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element (some indeterminacies remain for ). We use this to construct Hecke algebra actions on the cohomology of varieties associated to or its square, for groups of arbitrary rank. In a subsequent work, we construct actions associated to more general regular elements and we study their trace on cohomology.
Keywords
Cite
@article{arxiv.math/0410454,
title = {Cohomologie des vari\'{e}t\'{e}s de Deligne-Lusztig},
author = {François Digne and Jean Michel and Raphaël Rouquier},
journal= {arXiv preprint arXiv:math/0410454},
year = {2016}
}