Finite Length of Locally Analytic Vectors for $\mathrm{GL}_2(F)$ and $D_F^\times$

Abstract

Let $F/\mathbf Q_p$ and $E/\mathbf Q_p$ be finite extensions, and let $G=\mathrm{GL}_2(F)$ or $D_F^\times$, where $D_F$ is the quaternion division algebra over $F$. Let $Π$ be an admissible Banach $E$-representation of $G$ whose locally $\mathbf Q_p$-analytic vectors $Π^{\mathrm{an}}$ have an infinitesimal character. If $p>2$ or $F=\mathbf Q_p$, we prove that $Π^{\mathrm{an}}$ and $Π$ have finite length in their respective admissible categories.