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

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 $G = {\rm GL}_2({\mathbb Q}_p)$ associated to $\rho$ by the $p$-adic Langlands correspondence. We deduce from results due to Colmez, Dospinescu, Pa\v{s}k\={u}nas, Emerton, and others that if the locally analytic representation $\Pi(\rho)^\rm{la}$ has a subquotient isomorphic to a non-zero subquotient of a locally analytic principal series representation of $G$, then $\rho$ is trianguline.