Coxeter orbits and Brauer trees

We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive group G(F_q) when the order of q modulo l is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, L\"ubeck and Malle) and the derived equivalence predicted by the geometric version of Brou\'e's conjecture.