Constructing Weyl group multiple Dirichlet series

TL;DR

This paper develops a uniform method to construct Weyl group multiple Dirichlet series associated with root systems, demonstrating their meromorphic continuation and functional equations aligned with the Weyl group symmetries.

Contribution

It generalizes previous constructions to a broad class of root systems, establishing the existence and properties of these series uniformly.

Findings

Constructed Weyl group multiple Dirichlet series for various root systems.

Proved these series have meromorphic continuation to C^r.

Established the functional equations under Weyl group transformations.

Abstract

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties.