Exercise 6, § 2

Let G be a connected compact Lie group, T a maximal torus of $G,\mathfrak{g}$ (resp. $\mathfrak{t}$) the Lie algebra of G (resp. T), and $i:\mathfrak{t}\rightarrow \mathfrak{g}$ the canonical injection. Prove that the map $^ti:\mathfrak{g}^*\rightarrow \mathfrak{t}^*$ induces by passage to the quotient a homeomorphism from $\mathfrak{g}^*/G$ to $\mathfrak{t}^*/W_G(T)$.