Proposition 14

Let X be a subset of T, and let $R_X$ be the set of roots $\alpha \in R(G,T)$ such that $\alpha (X) =\{1\}$. The group $Z_G(X)/Z_G(X)_0$ is isomorphic to the quotient of the subgroup of $W_G(T)$ fixing X by the subgroup generated by the reflections $s_{\alpha}$ for $\alpha \in R_X$.

Put $H = Z_G(X)$; since $L(H)_{(\mathbf{C})}$ is the set of points of $\mathfrak{g}_{\mathbf{C}}$ fixed by Ad(X), it is the sum of $\mathfrak{t}_{\mathbf{C}}$ and the $\mathfrak{g}^{\alpha}$ for which $\alpha (X) =\{1\}$. Consequently, $R(H_0,T) =$ $R_X$, so $W_{H_0}(T)$ is generated by the reflections $s_{\alpha}$ for $\alpha \in R_X$. It now suffices to apply Prop. 7 of §2, no. 5.

We shall see below (§5, no. 3, Th. 1) that if G is simply-connected and X reduces to a point, the centralizer Z(X) is connected.