On the Irreducibility of Locally Analytic Vectors in Certain Banach Representations of $D^\times$

Abstract

We investigate the structure of locally analytic representations of $p$-adic groups, and establish a sufficient condition for the irreducibility of locally analytic vectors in certain Banach representations of $D^\times$, where $D$ is the quaternion division algebra over $\mathbb{Q}_p$. As an application, we prove that $\mathrm{JL}(ρ)^{\mathrm{an}}/\mathrm{JL}(ρ)^{1\text{-}\mathrm{alg}}$ is a topologically irreducible locally analytic representation of $D^\times$ under certain genericity conditions on $ρ$. Here $\mathrm{JL}(ρ)$ is the $ρ(-1)$-multiplicity space in the image of $Π(ρ)$ under Scholze's functor, and $Π(ρ)$ is the representation associated to $ρ$ by the $p$-adic local Langlands correspondence.