I posted this question here but I don't get any answer! I hope that someone could help me here.

Let G be a lie group with Lie algebra $\mathfrak{g}$. We denote by $S(\mathfrak{g}^*)$ the symmetric algebra of $\mathfrak{g}$. Let M be a smooth manifold on wich G acts. We denote by $\mathcal{A}(M)$ the space of differential forms on M.

There are two definitions of the space of G-equivariant differential forms on M:

the space of G-equivariant differential form on M is the space of polynomial maps $\alpha: \mathfrak{g} \rightarrow \mathcal{A}(M)$ such that $\alpha(gX)= g.\alpha(X)$ for $g \in G.$

The space of G-equivariant differential forms is ${(S(\mathfrak{g}^*) \otimes \mathcal{A}(M))}^G$ ,(where the coadjoint action of G on $\mathfrak{g}^*$ induced the G action on $S(\mathfrak{g}^*) $).

What is the explicit isomorphism between these two spaces ? I know that $S(\mathfrak{g}^*)$ can be identified with polynomial functions on $\mathfrak{g}$ and that the space $S(\mathfrak{g}^*) \otimes \mathcal{A}(M)$ is identified with the space $Hom_\mathbb{R}{((S(\mathfrak{g}^*))}^*, \mathcal{A}(M))$, however I couldn't write down precisely the relation between these two spaces!