Proposition 10

The map $u \rightarrow L(u)_{(\mathbf{C})}$ from Aut(T) to $\mathbf{G}\mathbf{L}(\mathfrak{t}_{\mathbf{C}})$ induces an isomorphism from W to the Weyl group of the split reductive Lie algebra $(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$. For all $\alpha \in R$, $W_{Z_{\alpha}}(T)$ is of order 2, and the image under the preceding isomorphism of the non-identity element of $W_{Z_{\alpha}}(T)$ is the reflection $s_{H_{\delta(\alpha)}}$.

The map under consideration is injective. It remains to show that its image is equal to $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$.

Let $g\in N_G(T)$. With the notations in Chap. VIII, §5, no. 2, we have Ad $g\in$ Aut($\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}}$)$\cap$ Int($\mathfrak{g}_{\mathbf{C}}$), so Ad $g\in$ Aut$_0(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}}) ($loc. cit., no. 5, Prop. 11). By loc. cit., no. 2, Prop. 4, the automorphism of $\mathfrak{t}_{\mathbf{C}}$ induced by Ad $g$ belongs to $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$. Thus, the image of W in $\mathbf{G}\mathbf{L}(\mathfrak{t}_{\mathbf{C}})$ is contained in $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$.

Let $\alpha \in R(G,T)$, and let $\nu :\mathbf{S}\mathbf{U}(2,\mathbf{C})\rightarrow G$ be a morphism of Lie groups having the properties in the Cor. of Th. 1. The image under $\nu$ of the element $\theta$ of $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ has the following properties (§3, no. 6, formulas (17)):

a) (Int$\nu (\theta )$)$(t) =t$ if $t\in$ Ker$\alpha$,

b) (Int$\nu (\theta )$)$(t) =t^{-1}$ if $t\in T\cap S_{\alpha}$.

It follows that Ad $\nu (\theta )$ induces the identity on Ker $\delta (\alpha )\subset \mathfrak{t}_{\mathbf{C}}$, and induces the map $x \rightarrow -x$ on $[\mathfrak{g}^{\alpha},\mathfrak{g}^{-\alpha}]$, hence coincides with the reflection $s_{H_{\delta(\alpha)}}$. Thus, the image of W contains all the $s_{H_{\delta(\alpha)}}$, and hence is equal to $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$. In particular $W_{Z_{\alpha}}(T)$ is of order 2, and hence consists of the identity and Int $\nu (\theta )$. This completes the proof of the proposition.