Corollary 1

Let A be an alcove of $\mathfrak{t},\overline{A}$ its closure, and $H_A$ the stabilizer of A in $W'_a$.

a) The group $W'_a$ is the semi-direct product of $H_A$ by $W_a$.

b) The exponential map $\overline{A}\rightarrow T$ and the canonical injection $T\rightarrow G$ induce by passage to the quotients and to subsets homeomorphisms

$\overline{A}/H_A\rightarrow T/W\rightarrow G/$Int(G)

$A/H_A\rightarrow T_r/W\rightarrow G_r/$Int(G).

Let $w'\in W'_a$; then $w'(A)$ is an alcove of $\mathfrak{t}$, and there exists (Prop. $2b$)$)$ a unique element $w$ of $W_a$ such that $w(A) =w'$(A), that is $w^{-1}w'\in H_A$. Since $W_a$ is normal in $W'_a$, this proves a).

The canonical injection of $\overline{A}$ into $\mathfrak{t}$ induces a continuous bijection $\theta :\overline{A}\rightarrow$ $\mathfrak{t}/W_a$ (Prop. $2c$)$)$, and this is a homeomorphism since A is compact. Since $W_a$ is normal in $W'_a$, the group $H_A$ operates canonically on $\mathfrak{t}/W_a($Algebra, Chap. I, §5, no. 4) and $\mathfrak{t}/W'_a$ can be identified with the quotient $(\mathfrak{t}/W_a)/H_A$; the map $\theta$ is compatible with the operations of $H_A$, hence induces by passage to the quotient a homeomorphism $A/H_A\rightarrow \mathfrak{t}/W'_a$. Further, exp$_{\Gamma}$ induces a homeomorphism from $\mathfrak{t}/\Gamma (T)$ to T, hence also a homeomorphism from $\mathfrak{t}/W'_a$ to $T/W$. Assertion b) follows from that and Cor. 1 of Prop. 5 of §2, no. 4.