Verifying local–global compatibility for automorphic Galois representations is a cornerstone of the reciprocity conjectures of Langlands, Fontaine--Mazur and Clozel and has many applications within the Langlands programme. The celebrated work of Harris–Lan–Taylor–Thorne and Scholze on attaching $\ell$-adic Galois representations to cohomological automorphic representations of $\mathrm{GL}_n$​ over CM fields makes it possible to address the problem beyond the self-dual case.

 

Starting with the breakthrough work of Allen–Calegari–Caraiani–Gee–Helm–Le Hung–Newton–Scholze–Taylor–Thorne, it is now known that local–global compatibility (up to semisimplification) holds in this level of generality under irreducibility and genericity assumptions on the residual representation. In this talk, I will discuss work on removing both of these assumptions, as well as an application to the vanishing of adjoint Bloch–Kato Selmer groups, generalising work of Newton–Thorne and A’Campo. This is joint work in progress with Lambert A’Campo, Jack Thorne, and Dmitri Whitmore.