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

We prove local-global compatibility results at $\mathrm{\ell }=p$ 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 $\mathrm{\ell }=p$ 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.