Proposition 1

Let $S^m(\mathfrak{h}^*)^W$ be the set of elements of $S^m(\mathfrak{h}^*)$ invariant under W. Then pr$_m(\psi (k[P]^W)) =\mathbf{S}^m(\mathfrak{h}^*)^W$.

It is clear from the preceding that pr$_m(\psi (k[P]^W))\subset \mathbf{S}^m(\mathfrak{h}^*)^W$. Every element of $\mathbf{S}^m(\mathfrak{h}^*)$ is a $k$-linear combination of elements of the form

pr$_m$(exp $\lambda$ ) $=$ (pr$_m\circ \psi$ )$(e^{\lambda})$

where $\lambda \in P$ (Lemma 1). Hence every element of $\mathbf{S}^m(\mathfrak{h}^*)^W$ is a linear combination of elements of the form

$\sum_{w\in W}w$((pr$_m\circ \psi$ )$(e^{\lambda})$) $=$ (pr$_m\circ \psi$ )$(\sum_{w\in W}w(e^{\lambda}))$,

each of which belongs to pr$_m(\psi (k[P]^W))$.