  Given a polytope $\sigma\subset \mathbb{R}^m$, its characteristic
distribution $\delta_\sigma$ generates a $D$-module which we call the
characteristic $D$-module of $\sigma$ and denote by $M_\sigma$. 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
certain 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.
