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Séminaire Lotharingien de Combinatoire, 91B.84 (2024), 12 pp.

# Claudia Alfes, Joshua Maglione and Christopher Voll

# Ehrhart Polynomials, Hecke Series, and Affine Buildings

**Abstract.**
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.

Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.

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