Asymptotic Euler-Maclaurin formula for Delzant polytopes

Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin-Sternberg [GS]. Then, the problem is to find a concrete formula for the each term of the expansion. In this paper, an asymptotic Euler-Maclaurin formula of the Riemann sums over Delzant polytopes is given. The formula given here is similar to the so-called local EulerMaclaurin formula of Berline-Vergne [BeV]. A concrete description of differential operators which appear in each term of the asymptotic expansion is given independently of the local Euler-Maclaurin formula. By using this formula, a concrete formula for each term in two dimension and a formula for the third term of the expansion in arbitrary dimension are given. Moreover, it is shown that the differential operators defined here coincide with homogeneous parts of the differential operators of infinite order defined in [BeV]. 0 Introduction In this paper, we consider asymptotics of the Riemann sums over lattice polytopes, RN (P ;φ) := 1 Nm ∑ γ∈(NP )∩Zm φ(γ/N), (0.1) where P is a lattice polytope in R and φ is a smooth function on P . Formulas for RN(P ;φ) with a polynomial φ, which is often called Euler-Maclaurin formula, are extensively investigated in combinatorics and toric geometry. If we take φ = 1, the Riemann sum RN (P ; 1) is ∗Research partially supported by JSPS Grant-in-Aid for Scientific Research (No. 217401117).