Equivariant formality in complex-oriented theories

Let G be a product of unitary groups and let (M, ) be a compact symplectic manifold with Hamiltonian G-action. We prove an equivariant formality result for any complex-oriented cohomology theory E^* (in particular, integral cohomology). This generalizes the celebrated result of Atiyah-Bott-Kirwan for rational cohomology. The proof does not use classical ideas but instead relies on a recent cohomological splitting result of Abouzaid-McLean-Smith for Hamiltonian fibrations over CP¹. Moreover, we establish analogues of the "localization" and "injectivity to fixed points" theorems for certain cohomology theories studied by Hopkins-Kuhn-Ravenel. As an application of these results, we establish a Goresky-Kottwitz-MacPherson theorem with Morava K-theory coefficients for Hamiltonian T-manifolds.