  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 $K_G(\pt)$ characterized by the property: $I_q\tau_p=\delta^q_p$. For $M$
a GKM manifold we give an explicit construction of these $\tau_p$'s in terms of
the associated GKM graph.
