Exercise 3, § 9

Let $k_1, . . . , k_l$ be the characteristic degrees of the algebra of invariants of W (cf. Chap. V, §5).

a) Show that, for all $j\geq 1$, the number of $i$ such that $k_i> j$ is equal to the number of $\alpha \in R_+$ such that $\langle \rho , H_{\alpha}\rangle =j$. (Reduce to the case in which R is irreducible, and use Chap. VI, §4, Exerc. $6c$)$.)$ (Cf. §5, Exerc. $5g$)$.)$

b) Deduce the formula

$\prod_{\alpha\in R_+}\langle \rho , H_{\alpha}\rangle =\prod_{i=1}^l(k_i-$ 1)!.