Let $M$ be a symplectic toric manifold acted on by a torus $\mathbb{ T}$. In this work we exhibit an explicit basis for the equivariant K-theory ring $\mathcal{ K}_{ \mathbb{ T}}(M)$ 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 $\mathcal{ K}_{ \mathbb{ T}}(M)$. 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 $H_{ \mathbb{ T}}(M; \mathbb{ Z})$ 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 Poincar\'e duals to the closures of unstable manifolds, and also the canonical classes introduced by Goldin and Tolman in [GT], which exist whenever the moment map is index increasing.
Categories: math.SG