Affine Weyl Group Multiple Dirichlet Series: Type $\tilde{A}$

TL;DR

This paper constructs a new class of multiple Dirichlet series associated with affine Weyl groups, extending finite cases, and proves their meromorphic continuation and symmetry properties over function fields.

Contribution

It introduces a novel construction of multiple Dirichlet series for affine Weyl groups, based on algebraic geometry axioms, with proven meromorphic continuation and symmetry.

Findings

Unique series satisfying four axioms is constructed.

Series exhibits meromorphic continuation to a large domain.

Series has the affine Weyl group as its symmetry group.

Abstract

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_q(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.