Weighted Ehrhart Theory and Orbifold Cohomology

We introduce the notion of a weighted $\delta$-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted $\delta$-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart $\delta$-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.