Exercise 4, § 2

Let G be a connected compact Lie group, T a maximal torus of G and S a sub-torus of T. Denote by $\Sigma$ (resp. F) the stabilizer (resp. the fixer) of S in $W_G(T)$.

a) Prove that the group $N_G(S)/Z_G(S)$ is isomorphic to the quotient $\Sigma /F$. b) Let H be a connected closed subgroup of G, such that S is a maximal torus of H. Show that every element of $W_H$(S), considered as an automorphism of S, is the restriction to S of an element of $\Sigma$.