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 $φ$ on $P$, and with Laurent polynomial weights $f_Q(y)\in \mathbb{Z}[y^{\pm 1}]$ associated to the faces $Q \preceq P$ of the polytope. For this purpose, we calculate equivariant $K$-theoretic Hodge-Chern classes of a torus-equivariant mixed Hodge module $\mathcal{M}$ on the toric variety $X_P$ associated to $P$. For any integer $\ell$, we introduce an equivariant Hodge $χ_y$-polynomial $χ_y(X_P, \ell D_P; [\mathcal{M}])$, with $D_P$ the corresponding ample Cartier divisor on $X_P$ (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 similar properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function given by Stanley's $g$-function of the polar polytope of $P$, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. More generally, motivated by the analogy to the Kazhdan-Lusztig theory, we introduce a duality involution on the free $ \mathbb{Z}[y^{\pm 1}]$-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion-Vergne in Ehrhart theory as well as the above-mentioned more recent combinatorial formula of Beck-Gunnells-Materov.