Vector partition functions and index of transversally elliptic operators

TL;DR

This paper connects the equivariant K-theory of certain G-manifolds with the space of functions satisfying cocircuit difference equations, providing a new framework for understanding the index of transversally elliptic operators.

Contribution

It identifies the equivariant K-theory with a space of functions satisfying cocircuit difference equations, linking index theory with vector partition functions.

Findings

Determined the equivariant K-theory of points with finite stabilizers.

Identified the range of the index map for G-transversally elliptic operators.

Proved the index map is an isomorphism on its image.

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 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.