@Tyler Lawson , I do not belive in complex $ \mathbb{Z} [ (G/H)^k ] $ . You have to have group structure on $G/H$ to explain what is "the same boundary operator".

@Yemon Choi, compactness is really important. But I think that assumption that group simply connected is not (correct me if I am mistaking).

Actually I am doubt whether my question makes sense. You see, the approach from statement 2 works only if Lie group is compact. But all finite dimensional representations of compact groups are direct sum of irreducible. And cohomology of irreducible not trivial representation is just 0. But it is still would be great to interpreter this 0. Or may be you can generalize my question for more general representations or groups?

I understand now. I even wrote comment 2 to make it less ambiguous. You see, I need an answer, analogous to statement 1 (about cohomology of a group). In that example "geometric interpretation" of representation of discrete group is local system.

I am asking about geometric interpretation of cohomology of Lie algebra. Not about geometric interpretation of representation itself. Sorry, but you answer some other question.

Yes, relative cohomology are cohomology of homogenious space! And moreover there is a spectral sequence of a bundle. But I want to get this sequence for arbitrary Lie algebra. Here have to be purely algebraic approach for this.

It is very interesting. Can you provide references?

math.ru.nl/~solleveld/scrip.pdf Here you can find a definition of relative Lie algebra cohomology. It is done by means of explicit complex but it is still a way to make sense of such kind of objects.

What you said is just that this approach does not work. But I mean something different. I even edited my question, wrote "analog of Lyndon–Hochschild–Serre spectral sequence".