  Let G be a torus acting linearly on a complex vector space M, and let X be
the list of weights of G in M. We determine the equivariant K-theory of the
open subset of M consisting of points with finite stabilizers. We identify it
to the space DM(X) of functions on the lattice of weights of G, satisfying the
cocircuit difference equations associated to X, introduced by Dahmen--Micchelli
in the context of the theory of splines in order to study vector partition
functions. This allows us to determine the range of the index map from
G-transversally elliptic operators on M to generalized functions on G and to
prove that the index map is an isomorphism on the image. This is a setting
studied by Atiyah-Singer which is in a sense universal for index computations.
