Residues of quadratic Weyl group multiple Dirichlet series

TL;DR

This paper derives explicit residue formulas for quadratic Weyl group multiple Dirichlet series, linking residues to root subsystems and providing new expressions for the associated averages, enhancing understanding of their structure.

Contribution

It introduces explicit residue formulas for quadratic Weyl group multiple Dirichlet series and connects these residues to root subsystem averages, offering new combinatorial insights.

Findings

Explicit residue formulas for the series

Residues expressed via orthogonal root subsystems

New combinatorial expressions for the averages

Abstract

We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system.