Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units

Abstract

We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest.