Example 1

Let G be a connected compact Lie group, operating on itself by inner automorphisms. The fixer of an element $x$ of G is simply the centralizer $Z(x)$ of $x$ in G; it contains every maximal torus containing $x$. It follows that the largest orbit type $\tau$ is the conjugacy class of the maximal tori of G. The open set $G_{(\tau)}$ is the set of very regular elements of G (§5, no. 1, Remark). Assume that G is simply-connected. Then $G_{(\tau)}$ is equal to the set $G_r$ of regular elements of G (§5, no. 2, Remark 2); if A is an alcove of a Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{g}= L(G)$, the composite map $\pi : A\longrightarrow^{exp}G_r\longrightarrow G_r/$Int(G) is an isomorphism of analytic manifolds. Indeed, this is a homeomorphism (§5, no. 2, Cor. of Prop. 2); let $a\in A$, put $t=$ exp $a$ and identify $T_t(G)$ with $\mathfrak{g}$ by means of the translation $\gamma (t)$. The tangent map $T_a(\pi )$ can then be identified with the composite of the canonical injection $\mathfrak{t}\rightarrow \mathfrak{g}$ and the quotient map $\mathfrak{g}\rightarrow \mathfrak{g}/$Im(Ad $t^{-1}-1$). Since $t$ is regular, $T_a(\pi )$ is an isomorphism, hence the stated result (Differentiable and Analytic Manifolds, Results, 5.7.8).