  In the geometric situation of the simple Shimura varieties of Kottwitz
studied in Harris and Taylor's book, we describe the monodromy filtration of
the vanishing cycles complex and the spectral sequence associated to it. We
prove in particular that this filtration coincides with the weight one up to
shift. Thanks to the Berkovich-Fargues' theorem, we deduce the description of
the local monodromy filtration of the Deligne-Carayol model. In application, we
obtain the description of the local components of cohomological automorphic
representations of certains unitary groups, a global Jacquet-Langlands
correspondence between two such groups and a proof of the weight-monodromy
conjecture for the cohomology of these Shimura varieties.
