On the Irreducibility of Locally Analytic Vectors in Certain Banach Representations of D^×

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^×, where D is the quaternion division algebra over Qₚ. As an application, we prove that JL(ρ)ᵃⁿ/JL(ρ)¹⁻ᵃˡᵍ is a topologically irreducible locally analytic representation of D^× under certain genericity conditions on $ρ$. Here 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.