I have some question about the equivariant differential forms on a smooth manifold: \ The equivariant differential forms over some smooth manifold $M$, on which the compact Lie group $G$ acts, are defined to be $$ \Omega_{G}^q(M)= \oplus_{2i+j=q} (S^{i}(g^{'}) \otimes \Omega^j(M))^G $$ where $g^{'}$ denotes the dual of the Lie algebra $\mathfrak g$ of $G$. Then many authors say that these forms can be considered as polynomial functions on the Lie algebra $\mathfrak g$ of $G$, but I am not sure how this is to be done. For example if we consider the element $(x_1 \otimes... \otimes x_i) \otimes \omega$ where $x_1,..., x_i$ are elements of $g^{'}$ and $\omega$ is a differential form, what is the evaluation of this element on some $a \in \mathfrak g$ in the Lie algebra of $G$.
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