Computation of generalized equivariant cohomologies of Kac–Moody flag varieties

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring HT*(X) can be described by combinatorial data obtained from its orbit decomposition. In this paper, we generalize their theorem in three different ways. First, our group G need not be a torus. Second, our space X is an equivariant stratified space, along with some additional hypotheses on the attaching maps. Third, and most important, we allow for generalized equivariant cohomology theories EG* instead of HT*. For these spaces, we give a combinatorial description of EG*(X) as a subring of ∏EG*(Fi), where the Fi are certain invariant subspaces of X. Our main examples are the flag varieties G/P of Kac–Moody groups G, with the action of the torus of G. In this context, the Fi are the T-fixed points and EG* is a T-equivariant complex oriented cohomology theory, such as HT*, KT* or MUT*. We detail several explicit examples.