Proposition 18

*Let $x \in V$. The following three properties are equivalent:

(i) $x \in \overline{C}$; (ii) $x \geq s_\alpha(x)$ for all $\alpha \in B(C)$ (with respect to the order relation defined by C); (iii) $x \geq w(x)$ for all $w \in W$.

Since $s_\alpha(x) = x - \langle x, \alpha^\vee \rangle \alpha$ and since $\overline{C}$ is the set of elements $x \in V$ such that $\langle x, \alpha^\vee \rangle \geq 0$ for all $\alpha \in B(C)$, the equivalence of (i) and (ii) is obvious. On the other hand, it is clear that (iii) $\Longrightarrow$ (ii). We show that (i) $\Longrightarrow$ (iii). Let $x \in \overline{C}$, and let $w \in W$. We argue by induction on the length $l(w)$ of $w$. The case $l(w) = 0$ is trivial. If $l(w) \geq 1$, $w$ can be written in the form $w = w' s_\alpha$, with $\alpha \in B(C)$ and $l(w') = l(w) - 1$. Then

The induction hypothesis shows that $x - w'(x)$ is positive. On the other hand,

Now $s_\alpha \in T_{w^{-1}}$, and Prop. 17 (ii) shows that $w(\alpha) < 0$. Hence the result.