Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the idea is to "average" left invariant forms on $G$ using a Haar measure.
Can we extend this result for non compact infinite dimensional manifolds? In particular, consider the based free loop space $\Omega_bM$ LM$ of a manifold $M$; this is an infinite dimensional $S^1$-manifold. Is there a way to compute the cohomology of $\Omega_bM$ LM$ using a model of "invariant forms" and the idea of averaging?
By a result of Chen, we know that iterated integrals of differential forms in $M$ yield the correct cohomology of $\Omega_bM$. LM$. Is this model related to Cartan's story of invariant forms?
These questions are a bit vague, but I guess how to make them precise is part of my question.
