Lifting trianguline Galois representations along isogenies

Abstract

Given a central isogeny $\pi\colon G\to H$ of connected reductive $\overline{\mathbb Q}_p$-groups, and a local Galois representation $\rho$ valued in $H(\overline{\mathbb Q}_p)$ that is trianguline in the sense of Daruvar, we study whether a lift of $\rho$ along $\pi$ is still trianguline. We give a positive answer under weak conditions on the Hodge--Tate--Sen weights of $\rho$, and the assumption that the trianguline parameter of $\rho$ can be lifted along $\pi$. This is an analogue of the results proved by Wintenberger, Conrad, Patrikis, and Hoang Duc for $p$-adic Hodge-theoretic properties of $\rho$. We describe a Tannakian framework for all such lifting problems, and we reinterpret the existence of a lift with prescribed local properties in terms of the simple connectedness of a certain pro-semisimple group. While applying this formalism to the case of trianguline representations, we extend a result of Berger and Di Matteo on triangulable tensor products of $B$-pairs.