Ehrhart polynomials, Hecke series, and affine buildings

Given a lattice polytope P and a prime p, we define a function from the set of primitive symplectic p-adic lattices to the rationals that extracts the ℓth coefficient of the Ehrhart polynomial of P relative to the given lattice. Inspired by work of Gunnells and Rodriguez Villegas in type A, we show that these functions are eigenfunctions of a suitably defined action of the spherical symplectic Hecke algebra. Although they depend significantly on the polytope P, their eigenvalues are independent of P and expressed as polynomials in p. We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat–Tits buildings associated with p-adic symplectic groups. We compute these zeta functions by enumerating p-adic lattices by their elementary divisors and, simultaneously, one Hermite parameter. We report on a general functional equation satisfied by these local zeta functions, confirming a conjecture of Vankov.