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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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Can you see why elements of $S^r(\mathfrak g)$ can be seen as homogeneous polynomial functions on~$\mathfrak g$ of degree $r$? – Mariano Suárez-Alvarez Jan 23 '12 at 3:36
The same question was originally posted on MathStackexchange, where it has received a comment by MattE, which is even more explicit than the one one by Mariano Suárez-Alvarez. – Giuseppe Jan 23 '12 at 8:15

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