Exercise 8, § 1

Let $R$ be a reduced root system, $C$ a chamber, $2\sigma$ the sum of the positive elements of $R^-$, $P'(R)$ the set of $x \in P(R)$ such that $\langle x, \sigma \rangle \in \mathbf{Z}$. Then, $Q(R) \subseteq P'(R) \subseteq P(R)$, and $P(R)/P'(R)$ is of order 1 or 2. Let $(\beta_1, \ldots, \beta_l)$ be the basis of $R^-$ corresponding to $C$, and put $2\sigma = n_1 \beta_1 + \cdots + n_l \beta_l$ where the $n_i$ are integers $> 0$. Then, $P(R) = P'(R)$ if and only if all the $n_i$ are even.