Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})

We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet–Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuil’s locally analytic socle conjecture for $\textrm {GL}_{n}(\mathbb {Q}_{p})$ and show the existence of companion points for simple reflections.