Morse theory on Hamiltonian G-spaces and equivariant K-theory

Let $G$ be a torus and $M$ a compact Hamiltonian $G$-manifold with finite fixed point set $M^G$. If $T$ is a circle subgroup of $G$ with $M^G=M^T$, the $T$-moment map is a Morse function. We will show that the associated Morse stratification of $M$ by unstable manifolds gives one a canonical basis of $K_G(M)$. A key ingredient in our proof is the notion of local index $I_p(a)$ for $a\in K_G(M)$ and $p\in M^G$. We will show that corresponding to this stratification there is a basis ${\tau }_p$, $p\in M^G$, for $K_G(M)$ as a module over $K_G(\text{pt})$ characterized by the property: $I_q{\tau }_p={\delta }_p^q$. For $M$ a GKM manifold we give an explicit construction of these ${\tau }_p$’s in terms of the associated GKM graph.