Computation of generalized equivariant cohomologies of Kac–Moody flag varieties
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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 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.