Some results on locally analytic socle for ${\mathrm{G}\mathrm{L}}_n({\mathbb{Q}}_p)$

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 ${\mathrm{G}\mathrm{L}}_n({\mathbb{Q}}_p)$.