@JohnKlein I'm considering the Cartan model. Nevertheless, any other reasonable version of equivariant cohomology is also good for me.

Thank you very much for your answer and would you like to explain a little bit about "the additive chain complex groups are not the place to study this as the singular complex given as a simplicial set enables the full structure to be exhibited explicitly."? Are additive chain complex and singular complex the same thing?

@QiaochuYuan May I ask what does "IIRC" stand for?

Do you emphasis on how to characterize "finite dimensional" in a categorical way?

I'm not sure what do you mean by "$K$-orbits" here. If $G$ is complex semisimple and $K$ is the maximal compact subgroup, it is well-known that the $K$-action on $G/B$ is also transitive, hence the $K$-orbit is $G/B$ itself. Maybe what you really mean is the isotropy group of the $K$-action.

Thank you very much for your answer! I think the tensor product you constructed using $U$ is a kind of "twisted tensor product", which is analogous to the twisted D-module in D-module theory, isn't it?