Exercise 24, § 1

We retain the notation and assumptions of the preceding exercise.

a) Let $p \in P(R)$, and let $W.p$ be the orbit of $p$ under $W$. Show that the following conditions are equivalent:

(i) $S(p) = W.p$.

(ii) $\langle p, \alpha^\vee \rangle = 0, 1$ or $-1$ for all $\alpha \in R$.

(If (ii) is not satisfied, construct an element $q \in S(p)$ such that $(q|q) < (p|p)$, and hence $q \notin W.p$.)

b) Let $X$ be a non-empty saturated subset of $P(R)$. Show that $X$ contains an element $p$ satisfying conditions (i) and (ii) above. (Take $p$ to be an element of $X$ of minimal length.)

c) Assume that $R$ is irreducible. Let $C$ be a chamber of $R$, let $B = \{\alpha_i\}_{i \in I}$ be the corresponding basis, and let

be the highest root of $R$. Let $J$ be the set of $i \in I$ such that $n_i = 1$. Let $p \neq 0$ be a dominant weight. Show that conditions (i) and (ii) of a) are equivalent to each of the following conditions:

² This exercise, hitherto unpublished, was communicated to us by R. Steinberg.

(iii) $\langle p, \gamma^* \rangle = 1$.

(iv) There exists $i \in J$ such that $p$ is equal to the corresponding fundamental weight.

A weight $p$ satisfying these conditions is called minuscule. Show that every non-empty saturated subset of $P(R)$ contains 0 or a minuscule weight.