  We study the geometry and the cohomology of the tamely ramified cover of
Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely
local way, most of a conjecture of Harris on the form of the $\ell$-adic
cohomologies, as well as the torsion freeness of the integral cohomology.
During the study, we obtain some results of independent interest on the
coefficient systems over the Bruhat-Tits building associated to $GL_d(K).$
