Correspondance locale de Langlands et monodromie des espaces de Drinfeld

Abstract

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..