  In this article, we study the De Rham cohomology of the first cover in the
Drinfel'd tower. In particular, we get a purely local proof that the
supercuspidal part realizes the local Jacquet-Langlands correspondence for
${\rm GL}_n$ by comparing it to the rigid cohomology of some Deligne-Lusztig
varieties. The representations obtained are analogous to the ones appearing in
the $\ell$-adic cohomology if we forget the action of the Weil group. The proof
relies on the generalization of an excision result of Grosse-Kl\"onne and on
the explicit description of the first cover as a cyclic cover obtained by the
author on a previous work.
