Remark 3

The special points of $W_a$ (Chap. VI, §2, no. 2) are the elements $x$ of $\mathfrak{t}$ such that $\delta (\alpha )(x)\in 2\pi i\mathbf{Z}$ for all $\alpha \in R(G,T) ($loc. cit., Prop. 3), that is such that exp $x\in C(G) ($§4, no. 4, Prop. 8). For such an element $x$ we have $wx-x\in N(G,T)$ for all $w\in W$ (Chap. VI, §1, no. 9, Prop. 27), so the stabilizers of $x$ in $W_a$ and $W'_a$ coincide. Let S be the set of special points of A; it follows from the preceding and Cor. 1 that the group $H_A$ operates freely on S, and that the exponential map induces a bijection from $S/H_A$ to C(G).