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We study some closed rigid subspaces of the eigenvarieties, constructed byusing the Jacquet-Emerton functor for parabolic non-Borel subgroups. As anapplication (and motivation), we prove some new results on Breuil's locallyanalytic socle conjecture for $\mathrm{GL}_n(\mathbb{Q}_p)$.

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