On the irreducibility of \(p\)-adic Banach principal series of \(p\)-adic \(\mathrm{GL}_3\)

Abstract

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