Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds

Let M be a symplectic toric manifold acted on by a torus T. In this work we exhibit an explicit basis for the equivariant K-theory ring K-T(M) which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of K-T(M). The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in [GK]. We apply the same techniques to exhibit an explicit basis for the equivariant cohomology ring H-T(M; Z) which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincare duals to certain smooth flow up submanifolds, and also the canonical classes introduced by Goldin and Tolman in [GT], which exist whenever the moment map is index increasing.