A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

Abstract

We study twisted Weyl group multiple Dirichlet series attached to symmetrizable Kac-Moody root systems, using the Chinta-Gunnells method to construct their $p$-parts. Our main result is a decomposition theorem for functions invariant under the twisted Chinta-Gunnells action: under natural analytic hypotheses, such a function has a unique expansion in terms of shifted Chinta-Gunnells averages, indexed by the dominant weights in the highest weight module determined by the twisting parameter. In particular, we show that this decomposition holds for twisted multiple Dirichlet series over rational function fields. For finite root systems, these results were proved by Friedlander. We also show that the relevant Chinta-Gunnells averages admit analytic continuation to the interior of the complexified Tits cone. In the affine $\widetilde{A}_1$ case, we prove extra functional equations, not arising from the Weyl group, for the untwisted average and for averages twisted by fundamental weights. As a consequence, we obtain an explicit formula for the multiple Dirichlet series with square-free twisting parameters, and show that it also satisfies an extra functional equation.