Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

Abstract

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space V , namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in V. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in V gives rise to an effectively computable SI-interpolator (and a local Euler-MacLaurin formula), an IS-interpolator (and a reverse local Euler-MacLaurin formula) and an IS 0 -interpolator (which interpolates between integrals and sums over interior lattice points.) Rigid complement maps can be constructed by choosing an inner product on V or by choosing a complete flag in V. The corresponding interpolators generalize and unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.