Exercise 20, § 1

Let $(\alpha_1, \ldots, \alpha_l)$ be a basis of an irreducible reduced root system $R$. Let $r$ and $p$ be two integers, with $p \geq 2$, such that:

a) Let $\alpha = c_1 \alpha_1 + \cdots + c_l \alpha_l$ be a root. Show that $(\alpha|\alpha) = (\alpha_1|\alpha_1)$ (in other words, that $\alpha$ is a long root) if and only if $p$ divides $c_{r+1} \ldots c_l$.

b) Let $h$ be the Coxeter number of $W(R)$. Show that the number of longest (resp. shortest) roots of $R$ is equal to $hr$ (resp. $h(l-r)$).