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

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.