Remark 2

Assume that the group G is semi-simple; denote by R (resp. $R^{\vee}$) the root system $R(G,T)$ (resp. $R^{\vee}(G,T)$), so that we have inclusions

The finite abelian groups $P(R)/Q(R)$ and $P(R^{\vee})/Q(R^{\vee})$ are in duality (Chap. VI, §1, no. 9); if $\widehat{M}$ denotes the dual group of the finite abelian group M, we deduce from the preceding canonical isomorphisms

In particular, the product of the orders of $\pi_1(G)$ and C(G) is equal to the connection index $f$ of $R(G,T) ($loc. cit.).

Now let $G'$ be another connected compact Lie group, $T'$ a maximal torus of $G'$. Let $f: G\rightarrow G'$ be an isomorphism of Lie groups such that $f(T) = T'$; denote by $f_T$ the isomorphism from T to $T'$ that it defines. Then $X(f_T)$ is an isomorphism from $D^*(G',T')$ to $D^*(G,T)$, denoted by $D^*(f)$, and $\Gamma (f_T)$ is an isomorphism from $D_*(G,T)$ to $D_*(G',T')$, denoted by $D_*(f)$. If $t\in T$, and if we put $g=f\circ$ Int $t=$ (Int $f(t)$)$\circ f$, then $D^*(g) = D^*(f), D_*(g) = D_*(f)$.