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 $\operatorname {SI}$-interpolator (and a local Euler-Maclaurin formula), an $\operatorname {IS}$-interpolator (and a reverse local Euler-Maclaurin formula) and an $\operatorname {IS}$-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.