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

TL;DR

This paper establishes an upper bound on the Gelfand-Kirillov dimension for certain admissible $p$-adic Banach representations of $ ext{GL}_2K$ and quaternion units, advancing understanding of their structure with infinitesimal characters.

Contribution

It proves a Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character, extending results to quaternion units and offering new insights in the theory of such representations.

Findings

Gelfand-Kirillov dimension $oxed{ ext{≤} [K: extbf{Q}_p]}$ for admissible representations

Extension of bounds to quaternion unit groups over $K$

New observations in $p$-adic Banach representation theory

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.