Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the normalizer of $T$ in $G$. The canonical apartment of $G$ should be the vector space $V = X_{\ast}(T) \otimes \mathbb R$, where $X_{\ast}(T)$ is the lattice of cocharacters of $T$.
There is a natural action of $N$ as a group of vector space automorphisms of $V$, induced from the action of $N$ on cocharacters of $T$:
$$ n. \gamma(x) = n\gamma(x)n^{-1} \tag{$n \in N, x \in k^{\ast}, \gamma \in X_{\ast}(T)$}$$ However, in Bruhat-Tits theory there should be a more general action of $N$ as a group of affine space automorphisms of $V$, satisfying the following properties:
The translations on $V$ should come from $T$, and the kernel of the action on $N$ should be $T_c$.
Since every affine space automorphism of $V$ is a linear transformation (the "linear part") followed by a translation, the linear part of the automorphism induced by any $n \in N$ should be the one I just described above (coming from the action $n.\gamma(x) = n\gamma(x)n^{-1}$).
The action should depend on some sort of choice of root vectors $x_{\alpha}: k \xrightarrow{\cong} U_{\alpha}$, where $\alpha$ is a root of $T$ in $G$, and $U_{\alpha}$ is the root subgroup of $\alpha$. The action should possibly depend on a choice of Borel subgroup, I'm not sure.
What should the action of $N$ on $V$ as a group of affine space automorphisms be? What is the cleanest way of defining it?
The choice of a Borel subgroup (that is, a set of simple roots $\Delta$) gives an action of $T$ on $V$ by translations. If $\varpi_{\alpha}^{\vee}$ is the coweight corresponding to a simple root $\alpha$, then to $t \in T$ we can associate the translation $$v \mapsto v + \sum\limits_{\alpha \in \Delta} \operatorname{ord}_k(\alpha(t)) \varpi_{\alpha}^{\vee}. \tag{$v \in V$}$$ If the normalizer $N$ split as a direct product $N/T \times T$, there is an easy way of defining the action of $N$, since the Weyl group $N/T$ already acts linearly on $V$. However, $N$ does not typically split in this way. I have some idea of how to define the action in general, although it depends on a choice of Borel subgroup $B = TU$ as well as a choice of "canonical Weyl group representatives" depending on a "splitting," i.e. choice of simple root vectors.
