  The conjecture stated by Carayol in [{\em Non-abelian Lubin-Tate theory.}
Automorphic forms, Shimura varieties and $L$-functions, vol II: 15--39,
Academic Press,1990] predicted that the {\em supercuspidal part} of the l-adic
cohomology of the moduli spaces classifying certain formal groups introduced by
Drinfeld would provide a simultaneaous "realization" of Langlands and
Jacquet-Langlands local correspondences for {\em supercuspidal} representations
of $GL_d$. By works of Harris, Boyer, Harris-Taylor and Hausberger, this
conjecture is now proved. On another hand, recent works by Boyer and Faltings
provide a description of all the cohomology spaces and show in particular that
these spaces do not retain enough information for Langlands correspondence of
non-supercupidal representations. From these works, our main aim is to show how
one can still get a simultaneous realization for {\em elliptic} representations
of $GL_d$, using the formalism of derived categories. More specifically, we
describe the action of the monodromy on the (suitably defined) cohomology
complex of the Drinfeld tower and get the lacking information back from this
description..
