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We describe the structure of the Whittaker or Gelfand-Graev module on a$n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori andspherical level. We express our answer in terms of the representation theory ofa quantum group at a root of unity attached to the Langlands dual group of $G$.To do so, we introduce an algebro-combinatorial model for these modules anddevelop for them a Kazhdan-Lusztig theory involving new generic parameters.These parameters can either be specialized to Gauss sums to recover the$p$-adic theory or to the natural grading parameter in the representationtheory of quantum groups. As an application of our results, we deduce geometricCasselman-Shalika type results for metaplectic covers, conjectured in aslightly different form by S. Lysenko, as well as prove a variant of G. Savin'slocal Shimura type correspondences at the Whittaker level.

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity