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 * ( F i ) , where the F i 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 F i 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.