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Arrangements of hyperplanes I: Rational functions and Jeffrey-Kirwan residue

Michel Brion, Michele Vergne

TL;DR

This work develops a rigorous algebraic framework for rational functions with poles on a hyperplane arrangement Δ by decomposing R_Δ into a free G_Δ and a torsion NG_Δ, and introducing the Jeffrey-Kirwan residue as a projection onto the OsΔ-structured part. It connects this algebraic picture to geometry and transform methods via the Laplace transform, giving explicit inverse transforms F^o_δ and a jump formula that links wall-crossing to residues along hyperplanes. By relating G_Δ to the Orlik-Solomon algebra A_Δ and by establishing dual bases through iterated residues, the authors construct a stratified Fourier transform that encodes the combinatorial and geometric data of Δ. The framework enables algebraic approaches to equivariant localization integrals and sets the stage for multidimensional Eisenstein series and further zeta-function results in subsequent work.

Abstract

Consider the space $R_Δ$ of rational functions of several variables with poles on a fixed arrangement $Δ$ of hyperplanes. We obtain a decomposition of $R_Δ$ as a module over the ring of differential operators with constant coefficients. We generalize to the space $R_Δ$ the notions of principal part and of residue, and we describe its relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of work by L. Jeffreys 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.

Arrangements of hyperplanes I: Rational functions and Jeffrey-Kirwan residue

TL;DR

This work develops a rigorous algebraic framework for rational functions with poles on a hyperplane arrangement Δ by decomposing R_Δ into a free G_Δ and a torsion NG_Δ, and introducing the Jeffrey-Kirwan residue as a projection onto the OsΔ-structured part. It connects this algebraic picture to geometry and transform methods via the Laplace transform, giving explicit inverse transforms F^o_δ and a jump formula that links wall-crossing to residues along hyperplanes. By relating G_Δ to the Orlik-Solomon algebra A_Δ and by establishing dual bases through iterated residues, the authors construct a stratified Fourier transform that encodes the combinatorial and geometric data of Δ. The framework enables algebraic approaches to equivariant localization integrals and sets the stage for multidimensional Eisenstein series and further zeta-function results in subsequent work.

Abstract

Consider the space of rational functions of several variables with poles on a fixed arrangement of hyperplanes. We obtain a decomposition of as a module over the ring of differential operators with constant coefficients. We generalize to the space the notions of principal part and of residue, and we describe its relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of work by L. Jeffreys 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.

Paper Structure

This paper contains 7 sections, 23 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Rational functions with poles on hyperplanes: Jeffrey-Kirwan residue
  3. Residue along a hyperplane
  4. Orlik-Solomon relations
  5. Laplace transform and Jeffrey-Kirwan residue
  6. The jump formula
  7. Orlik-Solomon relations and stratified Fourier transform

Key Result

Theorem 1

We have a direct sum decomposition of $S(V^*)$-modules Moreover, the space $G_{\Delta}$ is a free $S(V^*)$-module, and is freely generated by $S_{\Delta}$, while the space $NG_{\Delta}$ is the torsion submodule. Finally, $G_{\Delta}$ is the space of functions in $R_{\Delta}$ which vanish at infinity.

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Definition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 16 more