  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 $w\_0$ (some indeterminacies remain for $G\_2$). We use this to
construct Hecke algebra actions on the cohomology of varieties associated to
$w\_0$ 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.

Representation Theory;Group Theory