Let $G$ be a Lie group acting on a manifold $M$. 


Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by considering de Rham cohomology of simplicial manifold associated to it, denoted by $\mathcal{G}\_\bullet$.


Is there any relation between de Rham cohomology of Lie groupoid $\mathcal{G}$ (eg page 11 of Laurent-Gengoux–Tu–Xu, *Chern-Weil map for principal bundles over groupoids*, Math. Z. **255** (2007) pp451–491, arXiv:[math/0401420](https://arxiv.org/abs/math/0401420v3)) and the de Rham (??) cohomology of the Lie group $G$ and the de Rham cohomology of the manifold $M$?



De Rham cohomology of Lie groupoid
Proposition $13$ and Remark $16$ in page $10$ of [Cohomology or Stacks](https://www.math.ubc.ca/~behrend/CohSta-1.pdf) says that, there is a natural isomorphism 


$$H^i\_G(X)\rightarrow H^i(X\times G\rightrightarrows X)$$
where, $H^i\_G(X)$ is the $i^{\text{th}}$ equivariant cohomology of $X$ with respect to action of $G$ and $H^i(X\times G\rightrightarrows X)$ is the $i^{\text{th}}$ deRham cohomology of the transformation groupoid $(X\times G\rightrightarrows X)$.


