Corollary 2

— Let $m$ be an integer $\geqslant 0$. Every element of $\mathrm{S}^m(\mathfrak h^*)^W$ is a linear combination of polynomial functions on $\mathfrak h$ of the form $x\mapsto\mathrm{Tr}(\rho(x)^m)$, where $\rho$ is a finite dimensional linear representation of $\mathfrak g$.

By Prop. 1, $\mathrm{S}^m(\mathfrak h^*)^W=(\mathrm{pr}_m\circ\psi)(k[P]^W)$. Now $\mathbf{Z}[P]^W=\mathrm{ch}\,\mathscr{R}(\mathfrak g)$ (§7, no. 7, Th. 2 (ii)). Thus, by Chap. VI, §3, no. 4, Lemma 3, $\psi(k[P]^W)$ is the $k$-vector subspace of $\mathrm{S}(\mathfrak h)^*$ generated by $\psi(\mathrm{ch}\,\mathscr{R}(\mathfrak g))=\zeta(\mathrm{Tr}\,\mathscr{R}(\mathfrak g))$. Consequently, $\mathrm{S}^m(\mathfrak h^*)^W$ is the vector subspace of $\mathrm{S}^m(\mathfrak h^*)$ generated by $(\mathrm{pr}_m\circ\zeta\circ\mathrm{Tr})(\mathscr{R}(\mathfrak g))$. But, if $\rho$ is a finite dimensional linear representation of $\mathfrak g$,

for all $x\in\mathfrak h$.