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.
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asked Jul 30, 2019 at 22:31
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3$\begingroup$ 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. $\endgroup$– Igor BelegradekCommented Jul 31, 2019 at 2:20
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$\begingroup$ 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. $\endgroup$– Fofi KonstantopoulouCommented Jul 31, 2019 at 11:00
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$\begingroup$ 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. $\endgroup$– Igor BelegradekCommented Jul 31, 2019 at 11:43
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"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
