Galois irreducibility implies cohomology freeness for KHT Shimura varieties

Abstract

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 𝔽 ¯ ℓ -representation ρ 𝔪 ¯ is irreducible, under the hypothesis that [F(exp(2iπ/ℓ):F]>d, where F is the reflex field, d the dimension of the KHT Shimura variety and ℓ the residual characteristic.