Computation of generalized equivariant cohomologies of Kac-Moody flag varieties

Abstract. 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 H∗ T (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 E∗ G instead of H∗ T . For these spaces, we give a combinatorial description of E∗ G (X) as a subring of ∏ E∗ G (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 E∗ G is a T -equivariant complex oriented cohomology theory, such as H∗ T , K∗ T or MU∗ T . We detail several explicit examples.