L'espace sym\'etrique de Drinfeld et correspondance de Langlands locale I

Let $K$ be a finite extension of ${\mathbb{Q}}_p$. We study the geometry and cohomology of the tamely ramified cover of Drinfeld’s symmetric space over $K\mathrm{.}$ We prove, in a purely local way, that the cohomology in middle degree realises the local Langlands correspondence and local Jacquet-Langlands correspondence for the depth zero supercuspidal representations.