Proposition 15

Let $\varphi$ be an isomorphism from $D^*(G',T')$ to $D^*(G,T)$ (resp. from $D_*(G,T)$ to $D_*(G',T')$). There exists an isomorphism $f: G\rightarrow G'$ such that $f(T) = T'$ and $\varphi = D^*(f)$ (resp. $\varphi = D_*(f)$); if $f_1$ and $f_2$ are two such isomorphisms, there exists an element $t$ of T such that $f_2=f_1\circ$ Int $t$.

The second assertion follows immediately from Prop. 9 (no. 4); we prove the first for the covariant diagrams, for example. Denote by $\mathfrak{g}'$ (resp. $\mathfrak{t}'$) the Lie algebra of $G'$ (resp. $T'$), and by $\mathfrak{g}'_{\mathbf{C}}$ (resp. $\mathfrak{t}'_{\mathbf{C}}$) its complexified Lie algebra. By Chap. VIII, §4, no. 4, Th. 2 (i), there exists an isomorphism $\psi :\mathfrak{g}_{\mathbf{C}}\rightarrow \mathfrak{g}'_{\mathbf{C}}$ that maps $\mathfrak{t}_{\mathbf{C}}$ to $\mathfrak{t}'_{\mathbf{C}}$ and induces on $\Gamma (T)\subset \mathfrak{t}_{\mathbf{C}}$ the given isomorphism $\varphi :\Gamma (T)\rightarrow \Gamma (T')$. Then $\mathfrak{g}$ and $\psi^{-1}(\mathfrak{g}')$ are two compact forms of $\mathfrak{g}_{\mathbf{C}}$ that have the same intersection $\mathfrak{t}$ with $\mathfrak{t}_{\mathbf{C}}$; by §3, no. 2, Prop. 3, there exists an inner automorphism $\theta$ of $\mathfrak{g}_{\mathbf{C}}$ inducing the identity on $\mathfrak{t}_{\mathbf{C}}$ and such that $\theta (\mathfrak{g}) =\psi^{-1}(\mathfrak{g}')$. By replacing $\psi$ by $\psi \circ \theta$, we can assume that $\psi$ maps $\mathfrak{g}$ to $\mathfrak{g}'$. Further, by Prop. 4 of no. 2, there exists a unique morphism $f_T: T\rightarrow T'$ such that $\Gamma (f_T) =\varphi$. Then the restriction of $\psi$ to $\mathfrak{t}$ is $L(f_T)$, and by §2, no. 6, Prop. 8, there exists a unique morphism $f: G\rightarrow G'$ that induces $f_T$ on T and $\psi$ on $\mathfrak{g}_{\mathbf{C}}$. Applying the preceding to $\varphi^{-1}$ and $\psi^{-1}$ we obtain an inverse morphism to $f$, which is therefore an isomorphism. Then $D_*(f) =\Gamma (f_T) =\varphi$, hence the proposition.

Note that, if T and $T'$ are two maximal tori of G, the diagrams $D^*(G,T)$ and $D^*(G,T')$ are isomorphic (if $g\in G$ is such that $gTg^{-1}= T'$, then Int $g$ is an isomorphism from G to G that maps T to $T'$). Denote by $D^*(G)$ the isomorphism class of $D^*(G,T)$ (cf. Theory of Sets, Chap. II, §6, no. 2); this is a root diagram that depends only on G and is called the contravariant diagram of G. The covariant diagram $D_*(G)$ of G is defined similarly, and we obtain: