Exercise 23, § 1

Let $R$ be a reduced root system, and let $X$ be a subset of $P(R)$. Say that $X$ is saturated if the following condition is satisfied:

(S) For all $p \in X, \alpha \in R$ and $i \in \mathbf{Z}$ such that $i$ is between 0 and $\langle p, \alpha^\vee \rangle$, $p - i \alpha \in X$.

a) Show that every saturated subset is stable under the Weyl group $W$ of $R$.

b) Show that, for any subset $A$ of $P(R)$, there exists a smallest saturated subset $S(A)$ of $P(R)$ containing $A$.

c) Let $C$ be a chamber of $R$, and let $p \in P(R) \cap \overline{C}$ be a dominant weight. Let $S(p)$ be the smallest saturated subset of $P(R)$ containing $p$. Let $\Sigma(p)$ be the set of elements $p' \in P(R)$ such that

(i) $p \equiv p' \mod Q(R)$.

(ii) For all $w \in W$, $w(p') \leq w$ (with respect to the order relation defined by $C$).

Show that $\Sigma(p)$ is finite, contains $p$, and is saturated. Conclude that $S(p)$ is contained in $\Sigma(p)$.

d) Show that, if $\alpha$ is a longest element of $R$, $S(\alpha) = R \cup \{0\}$.