Weyl group multiple Dirichlet series of type C

Abstract

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. In type C, they conjecturally arise from the Fourier-Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold metaplectic cover of SO(2r+1). For any odd n, we construct an infinite family of Dirichlet series conjecturally satisfying the above analytic properties. The coefficients of these series are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg in [6] and [7] required n to be sufficiently large, so that coefficients could be described by Weyl group orbits. We demonstrate that our construction agrees with that of [6] and [7] in the case where both series are defined, and hence inherits the desired analytic properties for n sufficiently large. Moreover our construction is valid even for n=1, where we prove our series is a Whittaker coefficient of an Eisenstein series. This requires the Casselman-Shalika formula for unramified principal series and a remarkable deformation of the Weyl character formula of Hamel and King [20].