Towards the Finite Slope Part for GLn.

Let |$L$| be a finite extension of |${\mathbb{Q}}_p$| and |$n\geq 2$|⁠. We associate to a crystabelline |$n$| -dimensional representation of |${\operatorname{Gal}}(\overline L/L)$| satisfying mild genericity assumptions a finite length locally |${\mathbb{Q}}_p$| -analytic representation of |${\operatorname{GL}}_n(L)$|⁠. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [ 6 ], we prove that this representation indeed occurs in spaces of |$p$| -adic automorphic forms. We then use this latter result in the ordinary case to show that certain "ordinary" |$p$| -adic Banach space representations constructed in our previous work appear in spaces of |$p$| -adic automorphic forms. This gives strong new evidence to our previous conjecture in the |$p$| -adic case. [ABSTRACT FROM AUTHOR]