Towards the finite slope part for ${\mathrm{G}\mathrm{L}}_n$

Let $L$ be a finite extension of ${\mathbb{Q}}_p$ and $n\ge 2$. We associate to a crystabelline $n$-dimensional representation of $$\mathrm{G}\mathrm{a}\mathrm{l}$(\overline L/L)$satisfyingmildgenericityassumptionsafinitelengthlocally$$\mathbb{Q}$_p$-analyticrepresentationof$$\mathrm{G}\mathrm{L}$_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [BHS17a] 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.