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 $\prod E_G^{*}(F_i)$, where the $F_i$ are certain invariant subspaces of X. Our main examples are the flag varieties $\mathcal{G}/\mathcal{P}$ of Kac–Moody groups $\mathcal{G}$, with the action of the torus of $\mathcal{G}$. 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 ${\mathit{MU}}_T^{*}$. We detail several explicit examples.