Remark

For all $x\in \mathbf{R}$, we have

$($ exp(2$\pi ix$) 0 $)_{2\pi ix}$

$\nu$ 0 exp($-2\pi ix$)$=\nu_T(e$ ) = exp($xK_{\alpha}$). (14)

In particular:

$\nu (-0$1 $-01)=\nu_T(-1) =$ exp$(\frac{1}{2}K_{\alpha})$. (15)

It follows that $\nu$ is injective if and only if $K_{\alpha}\notin2\Gamma$(T), in other words if there exists $\lambda \in X(T)$ such that $\langle \lambda , K_{\alpha}\rangle \notin2\mathbf{Z}$. When $\mathfrak{g}_{\mathbf{C}}$ is simple, $\nu$ is injective unless $\mathfrak{g}_{\mathbf{C}}$ is of type $B_n, C(G) =\{1\}$ and $\alpha$ is a short root (cf. Chap. VI, Plates).

In the remainder of this paragraph we denote by $R^{\vee}(G,T)$ the set of nodal vectors $K_{\alpha}$ for $\alpha \in R(G,T)$. This is a subset of $\Gamma (T)$ that the canonical injection of $\Gamma (T)$ into $\mathfrak{t}_{\mathbf{C}}$ identifies with the homothety with ratio $2\pi i$ of the inverse root system $R^{\vee}(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}}) =\{H_{\delta(\alpha)}\}$ of $\delta (R)$. It follows that $R^{\vee}(G,T)$ generates the $\mathbf{R}$-vector space $L(T\cap D(G))$, and hence that its orthogonal complement in X(T) is $X(T/(T\cap D(G)))$.

Denote by Aut(T) the group of automorphisms of the Lie group T; the Weyl group $W = W_G(T) ($§2, no. 5) can be identified with a subgroup of Aut(T). On the other hand, recall (Chap. VIII, §2, no. 2, Remark 4) that the Weyl group $W(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$ of the split reductive algebra $(\mathfrak{g}_{\mathbf{C}},\mathfrak{t}_{\mathbf{C}})$ operates on $\mathfrak{t}_{\mathbf{C}}$, and thus is canonically identified with a subgroup of $\mathbf{G}\mathbf{L}(\mathfrak{t}_{\mathbf{C}})$.