Eigenvarieties over CM fields and trianguline representations

Abstract

We show that the Galois representations associated to points on certain (derived) eigenvarieties for $\operatorname{GL}_n$ over a CM field are trianguline with the expected Sen weights, verifying an analogue of a conjecture of Hansen in many cases. The proof follows the strategy of passing to a larger unitary group $\widetilde{G}$ of signature $(n,n)$, where the key new input is an analytic continuation result for an eigenvariety for $\widetilde{G}$ localised at an Eisenstein maximal ideal. We also discuss the (subtle) relation of eigenvarieties for $\operatorname{GL}_n$ with the trianguline variety.