  This article is the final one of a series of articles on certain blocks of
modular representations of finite groups of Lie type and the associated
geometry. We prove the conjecture of Brou\'e on derived equivalences induced by
the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter
elements whenever the defining characteristic is good. We also prove a
conjecture of Hi{\ss}, L\"ubeck and Malle on the Brauer trees of the
corresponding blocks. As a consequence, we determine the Brauer trees (in
particular, the decomposition matrix) of the principal $\ell$-block of $E_7(q)$
when $\ell \mid \Phi_{18}(q)$ and $E_8(q)$ when $\ell \mid \Phi_{18}(q)$ or
$\ell \mid \Phi_{30}(q)$.
