image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ is contained in a unique irreducible component of $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$, then $x$ is classical.

According to this theorem we need to understand the local geometry of the space $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$ at $x_v$. It turns out that much of this local geometry is controlled by the Weyl group element $w_{x,v}$ associated to $x$ which only depends on the image $x_v$ of $x$ in $X_{\rm tri}^\square({\overline{\rho}}_{x,v})$. For $v\in S_p$ denote by $\lg(w_{x,v})$ the length of the permutation $w_{x,v}$ and by $d_{x,v}$ the rank of the ${\mathbb{Z}}$-module generated by $w_{x,v}(\alpha)-\alpha$, as $\alpha$ ranges over the roots of $({\mathrm{Res}}_{F_v^+/{\mathbb{Q}}_p}{\mathrm{GL}}_n)\times_{{\mathbb{Q}}_p} \overline {\mathbb{Q}}_p\cong \prod_{\tau:\, F_v^+\hookrightarrow \overline{\mathbb{Q}}_p}{\mathrm{GL}}_n$. Then $d_{x,v}\leq \lg(w_{x,v})$, with equality if and only if $w_{x,v}$ is a product of distinct simple reflections (Lemma \[coxeter\]).

\[mainlocal\] Let $v\in S_p$ and let $X\subseteq X_{\rm tri}^\square(\overline \rho_{x,v})$ be a union of irreducible components that contain $x_v$ and satisfy the [*accumulation property*]{} of Definition \[accu\] at $x_v$. Then $$\dim T_{X,x_v}\leq \dim X+ \lg(w_{x,v})-d_{x,v}=\dim X_{\rm tri}^\square(\overline \rho_{x,v}) + \lg(w_{x,v})-d_{x,v},$$ where $T_{X,x_v}$ is the tangent space to $X$ at $x_v$. In particular $X$ is smooth at $x_v$ when $w_{x,v}$ is a product of distinct simple reflections.

The accumulation condition in Theorem \[mainlocal\] actually prevents 