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∈K
G
(M) and p∈M
G
. We will show that corresponding to this stratification there is a basis τ
p
, p∈M
G
, for K
G
(M) as a module over K
G
(pt) characterized by the property: I
q
τ
p
=δ
q
p
. For M a GKM manifold we give an explicit construction of these τ
p
's in terms of the associated GKM graph.