Proposition 2

a) The group $W_a$ (resp. $W'_a$) is the semi-direct product of W by $N(G,T)$ (resp. $\Gamma (T)$); the subgroup $W_a$ of $W'_a$ is normal.

b) The group W (resp. $W_a$) operates simply-transitively on the set of chambers (resp. alcoves ).

c) Let C be a chamber and A an alcove. Then $\overline{C}$ (resp. A, resp. A) is a fundamental domain for the operation of W on $\mathfrak{t}$ (resp. of $W_a$ on $\mathfrak{t}$, resp. of $W_a$ on $\mathfrak{t}$-- $\mathfrak{t}_r$). If $x\in \mathfrak{t}_r$ and $w\in W_a$ are such that $w(x) =x$, then $w=$ Id.

d) For every chamber C, there exists a unique alcove A such that $A\subset C$ and$\gamma \in \frac{0}{A}.\in \overline{A}$. For every alcove A, there exists a unique $\gamma \in N(G,T)$ such that

If $w\in W$ and $\gamma \in \Gamma$(T), then $wt_{\gamma}w^{-1}=t_{w(\gamma)}$ and $wt_{\gamma}w^{-1}t^{-1}_{\gamma}=t_{w(\gamma)-\gamma}$, with $w(\gamma )-\gamma \in N(G,T)$; this immediately implies a). The rest of the proposition follows from Chap. VI, §1, no. 5 and §2, nos. 1 and 2.