Symplectic Hecke eigenbases from Ehrhart polynomials

TL;DR

This paper demonstrates that Ehrhart polynomial coefficient functions form a basis of unimodular invariant valuations and are symplectic Hecke eigenfunctions in even dimensions, leading to new analytic and combinatorial insights.

Contribution

It establishes the symplectic Hecke eigenfunction property of Ehrhart coefficient functions in even dimensions and applies spherical function theory to derive new results.

Findings

Ehrhart coefficient functions form a basis of unimodular invariant valuations.

In even dimensions, these functions are symplectic Hecke eigenfunctions.

Derived new analytic, asymptotic, and combinatorial results for Ehrhart coefficients.

Abstract

For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about the arithmetic functions averaging $\ell$th Ehrhart coefficients.