Arrangement of hyperplanes. I: Rational functions and Jeffrey-Kirwan residue. (English) Zbl 0945.32003

Brion, Michel; Vergne, Michèle

Ann. Sci. Éc. Norm. Supér. (4) 32, No. 5, 715-741 (1999).

Abstract of the paper: “Consider the space \(R_{\Delta}\) of rational functions of several variables with poles on a fixed arrangement \(\Delta\) of hyperplanes. We obtain a decomposition of \(R_{\Delta}\) as a module over the ring of differential operators with constant coefficients. We generalize the notions of principal part and residue to the space \(R_{\Delta}\), and we describe their relations to Laplace transforms 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, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions”.

Reviewer: Jiancheng Zou (Beijing)

MSC:

32C22	Embedding of analytic spaces
52C35	Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Keywords:

arrangement of hyperplane; rational function; Jeffrey-Kirwan residue

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