Galois $overline{mathbb F}_l$-monodromy, level fixing and Iharas lemma

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 Iharas lemma.