Galois irreducibility implies cohomology freeness for KHT Shimura varieties

Abstract

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb{T}$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak{m}$ of $\mathbb{T}$, appear to be free. In this work, we obtain the same result for $\mathfrak{m}$ such that its associated Galois ${\overline{\mathbb{F}}}_l$-representation $\overline{{\rho }_{\mathfrak{m}}}$ 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.