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

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

## Abstract

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.

## Keywords

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

## References

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