# De Rham cohomology of homogeneous spaces

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.

• You can find a proof in Theorem 2.3 of [Chevalley-Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124]. The fulltext is available via google from where I am. Jul 31 '19 at 2:20
• Thanks for the reference. I checked the paper, but it seems Theorem 2.3 deals with Lie groups, not homogeneous spaces. Also, in this case it gives a correspondence between cohomology classes and equivariant forms. Is it it clear that such forms are in correspondence with $L$-invariant elements of $\Lambda^*$ in the homogeneous space. Jul 31 '19 at 11:00
• I got the reference from Nomizu's paper "On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups" who stidies the same problem for nilmanifolds. There on page 1 Nomizu says: "For the homogeneous spaces of compact Lie groups we have the well known theory of invariant integrals of E. Cartan, namely, the cohomology of a homogeneous space of a compact Lie group can be obtained from the complex of invariant differential forms on it" with reference to theorem 2.3. Jul 31 '19 at 11:43