Morris identities and the total residue for a system of type Ar

  • Velleda Baldoni-Silva
  • Michèle Vergne
Part of the Progress in Mathematics book series (PM, volume 220)


The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type A r = {(e i e j )|1 ≤ i, j ≤ (r+1), ij}. As pointed out by Zeilberger [Z], these calculations are mere reformulations of Morris identities [M], where the total residue function replaces here the iterated constant term. The proof we give of these identities follows closely (as suggested in [Z] Aomoto’s computation [Aom] of generalized Selberg integrals. Recall that Selberg [Se] proved that the following integral:
$$ {S_{r}}\left( {{k_{1}},{k_{2}},{k_{3}}} \right) = \int_{{{{[0,1]}^{r}}}} {\prod\limits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{(1 - {x_{i}})}^{{{k_{2}}}}}} } \prod\limits_{{1 \le i < j \le r}} {|({x_{i}} - {x_{j}}){|^{{{k_{3}}}}}dx} $$
is a product of Г functions. In this setting, k 1 , k 2 , k 3 are nonnegative integers. Here we will be interested in the Fourier transform of the function
$$ \phi ({k_{1}},{k_{2}},{k_{3}})({x_{0}},x) = \frac{1}{{\prod\nolimits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{({x_{0}} - {x_{i}})}^{{{k_{2}}}}}{{\prod\nolimits_{{1 \le i < j \le r}} {({x_{i}} - {x_{j}})} }^{{{k_{3}}}}}} }}, $$
more particularly on the value
$${{s}_{r}}({{k}_{1}},{{k}_{2}},{{k}_{3}})(\xi ) = \int_{{{{\mathbb{R}}^{{r + 1}}}}} {{{e}^{{i{{x}_{0}}\xi }}}\phi ({{k}_{1}},{{k}_{2}},{{k}_{3}})({{x}_{0}},x)d{{x}_{0}}dx}$$
where ϕ (k 1, k 2, k 3) (x 0, x) is interpreted as a boundary value of an holomorphic function. As shown by Jeffrey–Kirwan [JK], the value of the function s r (k 1, k 2, k 3) (ξ) is easily deduced from the knowledge of the total residue of the integrand. This reduces the problem to purely algebraic consideration: “integration” means that we will explicitly compute the function ϕ (k 1, k 2, k 3) (x 0, x) modulo derivatives in x 1, x 2..., x r according to the decomposition appearing in Equation 1.1.


Morris Identity Homogeneity Degree Total Residue Basic Subset Dimensional Real Vector Space 


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Velleda Baldoni-Silva
    • 1
  • Michèle Vergne
    • 2
  1. 1.Dipartimento di MatematicaUniversitá degli Studi di Roma Tor Vergata Via della Ricerca ScientificaRomaItaly
  2. 2.Centre de MathématiquesEcole PolytechniquePalaiseau CedexFrance

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