Exercise 7, § 8

Let $d_1, . . . , d_l$ be the characteristic degrees of the algebra $I(\mathfrak{g})$, cf. Chap. V, §5, no. 1. For any integer $n\geq 0$, denote by $r_n$ the number of elements of degree $n$ in a homogeneous basis of $\mathbf{S}(\mathfrak{g})$ over $I(\mathfrak{g})$ (cf. no. 3, Remark 2), and put

$r(T) =\sum_{n=0}^{\infty}r_nT^n$. Show that

$r(T) = (1-T)^{-N}\prod_{i=1}^l(1-T^{d_i})$, where N = dim($\mathfrak{g}$).\*