for instance in A. Parreau’s thesis [@Parreau].

Let us go back to the initial question.

\[q - non-simplicial\] Which geometry can be associated to a group ${\rm G}(k)$ when ${\rm G}$ is a reductive group over $k$, a (not necessarily discretely) valued field?

The answer to this question is a long definition to swallow, so we will provide some explanations immediately after stating it.

The starting point is again a $d$-dimensional Euclidean space, say $\Sigma_{\rm vect}$, together with a finite group $\overline W$ in the group of isometries ${\rm Isom}(\Sigma_{\rm vect}) \simeq {\rm O}_d({\bf R})$. By definition, a [*vectorial wall*]{} in $\Sigma_{\rm vect}$ is the fixed-point set in $\Sigma_{\rm vect}$ of a reflection in $\overline{W}$ and a [*vectorial Weyl chamber*]{} is a connected component of the complement of the union of the walls in $\Sigma_{\rm vect}$, so that Weyl chambers are simplicial cones.

Now assume that we are given an affine Euclidean space $\Sigma$ with underlying Euclidean vector space $\Sigma_{\rm vect}$. We have thus ${\rm Isom}(\Sigma) \simeq {\rm Isom}(\Sigma_{\rm vect}) \ltimes \Sigma_{\rm vect}\simeq {\rm O}_d({\bf R}) \ltimes {\bf R}^d$. We also assume that we are given a group $W$ of (affine) isometries in $\Sigma$ such that the vectorial part of 