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(Fq) 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, Lübeck and Malle (1995) in [25], ) and the derived equivalence predicted by the geometric version of Brouéʼs conjecture (Broué and Malle, 1993, [7]).