Exercise 13, § 7

Let $\lambda =\sum_{\alpha\in B}a_{\alpha}\alpha$ be an element of $P_{++}$. For $n= 0,1, . .$., denote by $\mathscr{X}_n$ the set of weights $\mu$ of $E(\lambda )$ such that $\lambda -\mu$ is the sum of $n$ elements of B. Let $s_n$ be the sum of the multiplicities of the elements of $\mathscr{X}_n$ (as weights of $E(\lambda )$). Let $T = 2\sum_{\alpha\in B}a_{\alpha}$. Show that:

$a) T$ is an integer $\geq 0$.

$b)s_n= 0$ for $n >T$, and $s_{T-n}=s_n$.

c) If $r$ is the integer part of $T/2$, then $s_1\leq s_2\leq \cdots \leq s_{r+1}$.