We tackle three related problems. The first deals with freeness of localized cohomology groups of Harris-Taylor perverse sheaves, defined on the special fiber of some Kottwitz-Harris-Taylor Shimura variety. We then study the nilpotent monodromy operator acting both on the global cohomology of KHT Shimura variety and on the perverse sheaf of vanishing cycles. We then exhibe cases of level fixing phenomenon in the sense where the level at some fixed place of any rise in characteristic zero of an irreducible automorphic representation, is fixed equals to the one modulo l.