  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.
