Remark 3

Denote by $\mathfrak{g}_r$ the inverse image of $G_r$ under the exponential map and by $\varepsilon :\mathfrak{g}_r\rightarrow G_r$ the map induced by exp$_G$. The map $(g, x) \rightarrow$ (Ad $g$)$(x)$ from $G\times \mathfrak{t}_r$ to $\mathfrak{g}_r$ defines by passage to the quotient a map $\psi_r: (G/T)\times \mathfrak{t}_r\rightarrow \mathfrak{g}_r$. We have $\varepsilon \circ \psi_r=\varphi_r$. Let $w\in W, \gamma \in \Gamma (T)$ and $\omega \in W'_a$ be such that $\omega (z) =w(z) +\gamma$ for all $z\in \mathfrak{t}$; then $\psi_r((\overline{g}, x)\omega ) =\psi_r(\overline{g}, x)-$ (Ad $g$)$(\gamma )$ for $g\in G, x\in \mathfrak{t}_r$, so $\psi_r((\overline{g}, x)\omega ) =\psi_r(\overline{g}, x)$ if and only if $\gamma = 0$. It follows (cf. General Topology, Chap. XI, in preparation) that $\psi_r$ is a principal covering of $\mathfrak{g}_r$ with group W, and that $\varepsilon :\mathfrak{g}_r\rightarrow G_r$ is a covering associated to the principal covering $\varphi_r$, with fibre isomorphic to the $W'_a$-set $W'_a/W$.