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The main purpose of this article is to extend some of the ideas from Schubertcalculus to the more general setting of Hamiltonian torus actions on compactsymplectic manifolds with isolated fixed points. Given a generic component ofthe moment map, we define a canonical class \alpha_p in the equivariantcohomology of the manifold M for each fixed point p of M. When they exist,canonical classes form a natural basis of the equivariant cohomology of M; inparticular, when M is a flag variety, these classes are the equivariantSchubert classes. We show that the restriction of a canonical class \alpha_p toa fixed point q can be calculated by a rational function which depends only onthe value of the moment map, and the restriction of other canonical classes topoints of index exactly two higher. Therefore, the structure constants can becalculated by a similar rational function. Our restriction formula ismanifestly positive in many cases, including when M is a flag manifold.Finally, we prove the existence of integral canonical classes in the case thatM is a GKM manifold the moment map component is index increasing. In this case,our restriction formula specializes to an easily computable rational sum whichdepends only on the GKM graph.

Towards Generalizing Schubert Calculus in the Symplectic Category