Galois $\overline{\mathbb F}_l$-monodromy, level fixing and Ihara's lemma
Résumé
We exhibit cases of a level fixing phenomenon for galoisian automorphic
representations of a CM field $F$, with dimension $d \geq 2$.
The proof rests on the freeness of the localized
cohomology groups of KHT Shimura varieties and the strictness of its
filtration induced by the spectral sequence associated to the filtration of
stratification of the nearby cycles perverse sheaf at some fixed place $v$ of $F$.
The main point is the observation that the action of the unipotent
monodromy operator at $v$ is then given by those
on the nearby cycles where its order of nilpotency modulo $l$ equals those in
characteristic zero. Finally we infer some consequences concerning level raising
and Ihara's lemma.
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