Exercise 9, § 2

This exercise is devoted to another proof of Th. 2. Let G be a connected compact Lie group, T a maximal torus of G, N its normalizer. Denote by $G\times^NT$ the quotient manifold of $G\times T$ by N for the operation defined by $(g, t).n= (gn, n^{-1}tn) ($Differentiable and Analytic Manifolds, Results, 6.5.1). a) Show that the morphism $(g, t) \rightarrow gtg^{-1}$ from $G\times T$ to G defines by passage to the quotient an analytic morphism $f: G\times^NT\rightarrow G$.

b) Let $\theta$ be an element of T whose set of powers is dense in T (General Topology, Chap. VII, §1, no. 3, Cor. 2 of Prop. 7). Show that $f^{-1}(\theta ) =\{x\}$, where $x$ is the class of $(e, \theta )$ in $G\times^NT$, and that $f$ is étale at $x$.

c) Conclude from Exerc. $7d)$ that $f$ is surjective, and deduce another proof of Th. 2.