Multiple Dirichlet series and Gauss sums

In this lecture I will discuss a family of multiple dirichlet series that are built out of sums of n-th order Gauss sums and a given root system of rank r. The combinatorics of the root system plays a key role in the definition. If n is sufficiently large, we show that these multiple Dirichlet series, originally defined in a product of r right half planes, have meromorphic continuation to C^r and satisfy a group of functional equations isomorphic to the Weyl group of the root system. If n is smaller, there are intriguing connections to combinatorics and representation theory in one case; these suggest a larger picture. This work is all joint with Brubaker and Bump, and parts are also joint with Chinta and Hoffstein.