Weighted Ehrhart theory via equivariant toric geometry

We give a K-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope P, depending on a given homogeneous polynomial function on P, and with Laurent polynomial weights fQ (y) Zy^ 1 associated to the faces Q P of the polytope. For this purpose, we calculate equivariant K-theoretic Hodge-Chern classes of an equivariant mixed Hodge module M on the toric variety XP associated to P (defined via an equivariant embedding of XP into an ambient smooth variety). For any integer, we introduce a corresponding equivariant Hodge ᵧ-polynomial ᵧ (XP, DP; M), with DP the corresponding ample Cartier divisor on XP (defined by the facet presentation of P). Motivic properties of the Hodge-Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying the corresponding properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function corresponding to Stanley's g-function of the polar polytope of P, we recover in geometric terms a recent combinatorial formula of Beck--Gunnells--Materov.