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

Cohomologie de de Rham du rev\^etement mod\'er\'e de l'espace de Drinfeld