Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincar\'e duality-type correspondence for equivariant K-theory.