Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$.
It is quite standard that the basic forms in $\Omega^\*(M) \otimes W(\mathfrak{g}^\*)$ form a model for the singular equivariant cohomology of $M$. However, I have never seen a proof and it is not straightforward to me. Could someone give a sketch or a reference of the proof of this fact? It is probably in one of Cartan's papers but I haven't been able to find it.




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Here goes some background:


We define its Weil algebra by $W^\*(\mathfrak{g}^\*)=S^\*(\mathfrak{g}^\*) \otimes \wedge^\*(\mathfrak{g}^\*)$ there is also a natural differential operator $d\_W$ which makes $W\*(\mathfrak{g}^\*)$ into a complex. We define $d\_W$ as follows:


Choose a basis $e\_1,...,e\_n$ for $\mathfrak{g}$ and let $e^\*\_1,...e^\*\_n$ its dual basis in $\mathfrak{g}^\*$. Let $\theta\_1,...,\theta\_n$ be the image of $e^\*\_1,...e^\*\_n$ in $\wedge(\mathfrak{g}^\*)$ and let $\Omega\_1,...,\Omega\_n$ be the image of $e^\*\_1,...e^\*\_n$ in $S(\mathfrak{g}^\*)$. Let $c\_{jk}^i$ be the structure constants of $\mathfrak{g}$, that is $[e\_j,e\_k]=\sum\_{i=1}^nc\_{jk}^ie\_i$. Define $d\_W$ by
\begin{eqnarray}
d\_W\theta\_i=\Omega\_i- \frac{1}{2}\sum\_{j,k} c\_{jk}^i \theta\_j \wedge \theta\_k
\end{eqnarray}
and
\begin{eqnarray}
d\_W\Omega\_i=\sum\_{j,k}c\_{jk}^i\theta\_j \Omega\_k
\end{eqnarray}
and extending $d\_W$ to $W(\mathfrak{g})$ as a derivation.


We can also define interior multiplication $i\_X$ on $W(\mathfrak{g}^\*)$ for any $X \in \mathfrak{g}$ by
\begin{eqnarray}
i\_{e\_r}(\theta\_s)=\delta^r\_s, i\_{e\_r}(\Omega\_s)=0
\end{eqnarray}
for all $r,s=1,...,n$ and extending by linearity and as a derivation. 


Now consider $\Omega^\*(M) \otimes W(\mathfrak{g}^\*)$ as a complex. Using this definition of interior multiplication, together with the usual definition of interior multiplication on forms, we define the basic complex of $\Omega^\*(M) \otimes W(\mathfrak g^\*)$:


We call $\alpha \in \Omega^\*(M) \otimes W(\mathfrak{g}^\*)$ a basic element if $i\_X(\alpha)=0$ and $i\_X(d \alpha)=0$. Basic elements in $\Omega^\*(M) \otimes W(\mathfrak{g}^\*)$ form a subcomplex which we denote by $\Omega^\*\_G (M)$.


The claim is that $H^\*(\Omega^\*\_G (M))=H^\*(M \times\_G EG)$ where the right hand side denotes the singular equivariant cohomology of $M$.



Cartan-Weil model for Equivariant Cohomology
see the very nice book of [Guillemin-Sternberg (Supersymmetry and ...)](http://books.google.com/books?id=zYMp0GWLFiAC&lpg=PA248&ots=Bx2FxpUDmI&dq=guillemin%2520sternberg%2520supersymmetry&pg=PA182#v=onepage&q&f=false); it also has a reprint of Cartan's paper.


