Trianguline representations and locally analytic principal series of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$

Abstract

Let $\rho$ be an absolutely irreducible 2-dimensional $p$-adic representation of the absolute Galois group of ${\mathbb{Q}}_p$, and let $\Pi (\rho )$ be the unitary Banach space representation of $G={\mathrm{GL}}_2({\mathbb{Q}}_p)$ associated to $\rho$ by the $p$-adic Langlands correspondence. We deduce from results due to Colmez, Dospinescu, Paškūnas, Emerton, and others that if the locally analytic representation $Π(ρ)^\rm{la}$ has a subquotient isomorphic to a non-zero subquotient of a locally analytic principal series representation of $G$, then $\rho$ is trianguline.