Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a corresponding inducing $L$-module $\Lambda^*$. Knowing that every cohomology class has a $G$-covariant representative, it is not too difficult to see that the cohomology classes are in bijective correspondence with the left $L$-invariant elements of $\Lambda^*$. This fact is surely very well known, but I can't find a reference in the literature. Can somebody point me to the right place.
"The averaging process" which is used in [Chevalley-Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124] on pages 90-92, may be used not only for compact Lie groups, but almost literally in the same form for any homogeneous spaces of such Lie groups.
In more general situation (for arbitrary linear representations) the proof may be found here (see chapter 1, paragraph 4): A. L. Onishchik, Topology of Transitive Transformation Groups. 1994