Weyl group multiple Dirichlet series of type C

TL;DR

This paper develops the theory of Weyl group multiple Dirichlet series for root systems of type C, constructing new series with analytic continuation and functional equations related to the Weyl group, including cases with minimal assumptions on n.

Contribution

It introduces a novel construction of Weyl group multiple Dirichlet series of type C valid for all positive integers n, including n=1, extending previous work that required larger n.

Findings

Constructed Dirichlet series for all odd n, including n=1.

Proved the series are Whittaker coefficients of Eisenstein series for n=1.

Demonstrated agreement with previous constructions for large n.

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].