Multiple Dirichlet Series for Affine Weyl Groups

2014/06/03 by Ian Whitehead, Whitehead, Ian
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT

paper · pdf · doi:10.48550/arxiv.1406.0573

arxiv created 2014/06/03 · arxiv updated 2014/06/04

Abstract

Let W be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting A_n for n even. We construct a multiple Dirichlet series Z(x₁, … x_n+1), meromorphic in a half-space, satisfying a group W of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this way is optimal for computing moments of character sums and L-functions, including the fourth moment of quadratic L-functions at the central point via D₄ and the second moment weighted by the number of divisors of the conductor via A₃. We also give evidence to suggest that this series appears as a first Fourier-Whittaker coefficient in an Eisenstein series on the twofold metaplectic cover of the relevant Kac-Moody group. The construction is limited to the rational function field \mathbbF_q(t), but it also describes the p-part of the multiple Dirichlet series over an arbitrary global field.

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