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 FourierWhittaker coefficients of minimal parabolic Eisenstein series on an nfold 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 GelfandTsetlin 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 CasselmanShalika formula for unramified principal series and a remarkable deformation of the Weyl character formula of Hamel and King [20].
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.1158
 Bibcode:
 2010arXiv1003.1158B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11F68;
 05E10
 EPrint:
 contains minor revisions