Cohomology of Deligne-Lusztig varieties for short-length regular elements in exceptional groups. (Q2438374)

The author computes the \(\ell\)-adic cohomology for a few specific Deligne-Lusztig varieties for split groups of type \(F_4\) or \(E_n\). The results serve as test cases for the Abelian defect conjecture of Broué. As a by-product the author obtains a conjecture for the Brauer trees of the principal generic \(\Phi_{14}\)-block of \(E_7\) and the principal generic \(\Phi_{24}\)-block of \(E_8\).