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Suppose that $G$ is the group of $F$-points of a connected reductive groupover $F$, where $F/\mathbb{Q}_p$ is a finite extension. We study the(topological) irreducibility of principal series of $G$ on $p$-adic Banachspaces. For unitary inducing representations we obtain an optimalirreducibility criterion, and for $G = \mathrm{GL}_n(F)$ (as well as forarbitrary split groups under slightly stronger conditions) we obtain a variantof Schneider's conjecture [Sch06, Conjecture 2.5]. In general we reduce theirreducibility problem to smooth inducing representations and almost simplesimply-connected $G$. Our methods include locally analytic representationtheory, the bifunctor of Orlik--Strauch, translation functors, as well as newresults on reducibility points of smooth parabolic inductions.

On the irreducibility of $p$-adic Banach principal series of $p$-adic reductive groups