Ordinary parts and local-global compatibility at $\ell=p$

Abstract: We prove local-global compatibility results at $\ell=p$ for the torsion automorphic Galois representations constructed by Scholze, generalising the work of Caraiani--Newton. In particular, we verify, up to a nilpotent ideal, the local-global compatibility conjecture at $\ell=p$ of Gee--Newton in the case of imaginary CM fields under some technical assumptions. The key new ingredient is a local-global compatibility result for $Q$-ordinary self-dual automorphic representations for arbitrary parabolic subgroups.