Vector partition functions and index of transversally elliptic operators

Abstract

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 topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice \( \widehat{G} \), satisfying the cocircuit difference equations associated to X, introduced by Dahmen and Micchelli in the context of the theory of splines in order to study vector partition functions (cf. [7]).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 and Singer [1] which is in a sense universal for index computations.