  Let $L$ be a finite extension of $\mathbb{Q}_p$ and $n\geq 2$. We associate
to a crystabelline $n$-dimensional representation of $\mathrm{Gal}(\overline
L/L)$ satisfying mild genericity assumptions a finite length locally
$\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$. 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.
