trianguline representations ([@Co]).

In the present paper we prove new cases of this classicality conjecture (in the above unitary setting). In particular we are able to deal with cases where the overconvergent form $x$ is [*critical*]{}. Throughout, we assume that $\rho_x$ is crystalline at $p$-adic places. Essentially the same proof should work if $\rho_x$ is only assumed crystabelline, but the crystalline assumption significantly simplifies the notation.

To state our main results, we fix an overconvergent form $x$ of classical weight such that $\rho_x$ is crystalline at all places dividing $p$. Such an overconvergent form can be described by a pair $(\rho_x,\delta_x)$, where $\rho_x$ is as above and $\delta_x=(\delta_{x,v})_{v\in S_p}$ is a locally ${\mathbb{Q}}_p$-analytic character of the diagonal torus of $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)\cong \prod_{v\in S_p}{\mathrm{GL}}_n(F^+_v).$ Here $S_p$ denotes the set of places of $F^+$ dividing $p$. There are nontrivial relations between $\rho_{x,v}:=\rho\vert_{{\rm Gal}(\overline {F^+_v}/F^+_v)}$ and $\delta_{x,v}$, in particular the character $\delta_{x,v}$ defines an ordering of the eigenvalues of the crystalline Frobenius on $D_{\rm cris}(\rho_{x,v})$. If we assume that these Frobenius eigenvalues are pairwise distinct, then this ordering defines a Frobenius stable flag in $D_{\rm cris}(\rho_{x,v})$. We can therefore associate to $x$ for each $v\in S_p$ a permutation $w_{x,v}$ that gives the relative position 