  Harris and Taylor proved that the supercuspidal part of the cohomology of the
Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands
correspondences, as conjectured by Carayol. Recently, Boyer computed the
remaining part of the cohomology and exhibited two defects : first, the
representations of GL\_d which appear are of a very particular and restrictive
form ; second, the Langlands correspondence is not realized anymore. In this
paper, we study the cohomology complex in a suitable equivariant derived
category, and show how it encodes Langlands correspondance for all elliptic
representations. Then we transfer this result to the Drinfeld tower via an
enhancement of a theorem of Faltings due to Fargues. We deduce that Deligne's
weight-monodromy conjecture is true for varieties uniformized by Drinfeld's
coverings of his symmetric spaces.
