On the irreducibility of $p$-adic Banach principal series of $p$-adic ${\mathrm{G}\mathrm{L}}_3$

We establish an optimal (topological) irreducibility criterion for $p$-adic Banach principal series of ${\mathrm{G}\mathrm{L}}_n(F)$, where $F\mathrm{/}{\mathbb{Q}}_p$ is finite and $n\le 3$. This is new for $n=3$ as well as for $n=2$, $F\mathrm{\ne }{\mathbb{Q}}_p$ and establishes a refined version of Schneider’s conjecture [Sch06, Conjecture 2.5] for these groups.