B-splines,Polytopes, and Their Characteristic D-modules

Given a polytope σ ? ? ^m , its characteristic distribution δ_σ generates a D-module which we call the characteristic D-module of σ and denote by M _σ. More generally, the characteristic distributions of a cell complex K with polyhedral cells generate a D-module M _K , which we call the characteristic D-module of the cell complex. We prove various basic properties of M _K , and show that under mild topological conditions on K, the D-module theoretic direct image of M _K coincides with the module generated by the B-splines associated to the cells of K (considered as distributions). We also give techniques for computing D-annihilator ideals of polytopes.