On the potential automorphy and the local-global compatibility for the monodromy operators at $p\mathrm{\ne }l$ over CM fields

Let $F$ be a CM field. In this paper, we prove the local-global compatibility for cohomological cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_n({\mathbb{A}}_F)$ at $p\mathrm{\ne }l$ by using certain potential automorphy theorems in some cases including higher dimensional cases. Moreover, we also prove the Ramanujan conjecture for cohomological cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_F)$.