On the Irreducibility of p-Adic Banach Principal Series of p-Adic \(\textrm{GL}_3\)

We establish an optimal (topological) irreducibility criterion for p-adic Banach principal series of \(\textrm{GL}_{n}(F)\) ${\text{GL}}_n(F)$ , where \(F/\mathbb {Q}_p\) $F/{\mathbb{Q}}_p$ is finite and \(n \le 3\) $n\le 3$ . This is new for \(n = 3\) $n=3$ as well as for \(n = 2\) $n=2$ , \(F \ne \mathbb {Q}_p\) $F\ne {\mathbb{Q}}_p$ and establishes a refined version of Schneider’s conjecture (Schneider, P.: International congress of Mathematics, vol. II, pp. 1261–1282, 2006, Conjecture 2.5) for these groups.