Constructing Weyl group multiple Dirichlet series

Abstract

Let Φ \Phi be a reduced root system of rank r r . A Weyl group multiple Dirichlet series for Φ \Phi is a Dirichlet series in r r complex variables s 1 , … , s r s_1,\dots ,s_r , initially converging for R e ( s i ) \mathrm {Re}(s_i) sufficiently large, that has meromorphic continuation to C r {\mathbb C}^r and satisfies functional equations under the transformations of C r {\mathbb C}^r corresponding to the Weyl group of Φ \Phi . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.