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 paper 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 galoisian $\overline{\mathbb F}_l$-representation $\overline{\rho_{\mathfrak m}}$ is irreducible.