On the p-parts of quadratic Weyl group multiple Dirichlet series

Abstract

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s₁,…, s_r, initially converging for ℜ(s_i) sufficiently large, which has meromorphic continuation to ℂ^r and satisfies functional equations under the transformations of ℂ^r corresponding to the Weyl group of Φ. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91–114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1–26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325–355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327–353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Φ = A_r, the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable A_r, Ann. Math. 166 (2007), 293–316.] when n = 2.