I have difficulties to understand the connection between equivariant and basic cohomology. I understand the definition of them but not how they are related (the Weil algebra killed me :)).
For example, consider a smooth manifold $M$ with a smooth action of a Lie group $G$. If we assume all orbits have the same dimension, this induces a foliated structure on M whose leaves are the orbites of the action. So, we have two cohomologies to look at: the basic and the equivariant ones. How are they related?
I have no special assumptions on $G$ and $M$. $G$ has no reason to be compact, and the action is not even proper (for example, the foliation of $S^1\times S^1$ induced by the foliation of $[0,1]\times[0,1]$ by parallel lines of non rational slope is a case I would like to understand). In the special case I am looking, $G$ is commutative but I am also interested in what happens when $G$ is not.
Hope my question is not too trivial. If it is, please give me a good reference on the subject (with a differential geometer point of view if possible).
Thanks in advance.