Koszul Duality for modules over Lie algebra

Let G be a compact Lie group. Set Λ• = H∗(G) and S • = H(BG). The coefficients are in R or C. Suppose G acts on a reasonable space X. In the paper [GKM] Goresky, Kottwitz and MacPherson established a duality between the ordinary cohomology which is a module over Λ• and equivariant cohomology which is a module over S • . This duality is on the level of chains, not on the level of cohomology. The Koszul duality says that there is an equivalence of derived categories of Λ•–modules and S • –modules. One can lift the structure of S • –module on H G(X) and the structure of Λ•–module on H(X) to the level of chains in such a way that the obtained complexes correspond to each other under the Koszul duality. The equivariant coefficients in the sense of [BL] are also allowed. Later, Allday and Puppe ([AP])