Arrangement of hyperplanes. I: Rational functions and Jeffrey-Kirwan residue

Consider the space R-Delta of rational functions of Several variables with poles on a fixed arrangement Delta of hyperplanes. We obtain a decompositio n of R-Delta as a module over the ring of differential operators with const ant coefficients. We generalize the notions of principal part and of residu e to the space R-Delta, and we describe their relations to Laplace transfor ms of locally polynomial functions. This explains algebraic aspects of the work by L. Jeffrey and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, an d we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions. (C) Elsevier, Paris.