Weyl Group Multiple Dirichlet Series I

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are