GALOIS IRREDUCIBILITY IMPLIES COHOMOLOGY FREENESS FOR KHT SHIMURA VARIETIES

Given a KHT Shimura variety with an action of its unramified Hecke algebra T , we proved in [7], see also [12] for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal m of T , happen to be free. In this work, we obtain the same result for m such that its associated Galois Fl-representation rho m is irreducible, under the hypothesis that [F(exp(2i pi/l) : F] > d , where F is the reflex field, d the dimension of the KHT Shimura variety and l the residual characteristic.