Proposition 4

a) Let $g\in G, t\in T, x\in \mathfrak{t}$, and let $\overline{g}$ be the image of $g$ in $G/T$. The following conditions are equivalent:

(i) $t\in T_r$ (resp. $x\in \mathfrak{t}_r$).

(i bis) The element $f(\overline{g}, t)$ (resp. $\varphi (\overline{g}, x)$) is regular in G.

(ii) The map $f$ (resp. $\varphi$ ) is a submersion at the point $(\overline{g}, t)$ (resp. $(\overline{g}, x)$).

(ii bis) The map $f$ (resp. $\varphi$ ) is étale at the point $(\overline{g}, t)$ (resp. $(\overline{g}, x)$).

b) The map $f_r$ (resp. $\varphi_r$, resp. $\varphi_A$) makes $(G/T)\times T_r$ (resp. $(G/T)\times \mathfrak{t}_r$, resp. $(G/T)\times A$) a principal covering of $G_r$ with group W (resp. $W'_a$, resp. $H_A$).

a) The equivalence of (i) and (i bis) is clear; that of (ii) and (ii bis) follows from the relations dim((G$/T$)$\times T$) $=$ dim((G$/T$)$\times \mathfrak{t}$) $=$ dim(G). By Lemma 4, $f$ is a submersion at the point $(\overline{g}, t)$ if and only if $\mathfrak{g}=\mathfrak{t}+$ Im(Ad $t-$Id), which means that $t$ is regular. Finally, since $\varphi =f\circ$ (Id$_{G/T}\times$ exp$_T$)$,\varphi$ is étale at the point $(\overline{g}, x)$ if and only if $f$ is étale at the point $(\overline{g}$, exp $x$), which by the preceding means that $x$ belongs to $\mathfrak{t}_r$.

b) The morphisms $f_r, \varphi_r, \varphi_A$ are thus étale. On the other hand, W operates freely on $G/T$, and a fortiori on $(G/T)\times T$. Let $g, g'$ in G and $t, t'$ in $T_r$ be such that $f(\overline{g}, t) =f(\overline{g}', t')$; then Int $g^{-1}g'$ maps $t'$ to $t$, and hence normalizes T, since $T = Z(t)_0= Z(t')_0$, and the class $w$ of $g^{-1}g'$ in W maps $(\overline{g}, t)$ to $(\overline{g}', t')$. It follows that $f_r$ is a principal covering with group W; this immediately implies that $\varphi_r$ is a principal covering with group $W'_a$, and hence by restriction to the connected component $(G/T)\times A$ of $(G/T)\times \mathfrak{t}_r$, that $\varphi_A$ is a principal covering with group $H_A$.