Equivariant formality in complex-oriented theories

Let $G$ be a product of unitary groups and let $(M,\omega)$ be a compact symplectic manifold with Hamiltonian $G$-action. We prove an equivariant formality result for any complex-oriented cohomology theory $\mathbb{E}^*$ (in particular, integral cohomology). This generalizes the celebrated result of Atiyah-Bott-Kirwan for rational cohomology from the 1980s. The proof does not use classical ideas but instead relies on a recent cohomological splitting result of Abouzaid-McLean-Smith for Hamiltonian fibrations over $\mathbb{CP}^1.$ 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.