A metaplectic Casselman-Shalika formula for GL r

Source abstract

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in the complex parameter. In addition we relate a specific spherical Whittaker function to the $p$-power part of the Weyl group multiple Dirichlet series for the root system of type $A_{r-1}$ constructed from $n$th order Gauss sums. We also show that the zonal spherical functions can be computed explicitly in terms of Hall-Littlewood polynomials as in Macdonald's formula for ${\rm GL}_r$.