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

TL;DR

This paper explores the structure of Whittaker modules on metaplectic covers of p-adic groups, linking them to quantum group representations at roots of unity, and introduces a combinatorial model with a Kazhdan-Lusztig theory.

Contribution

It introduces an algebro-combinatorial model and a Kazhdan-Lusztig theory for Whittaker modules on metaplectic covers, connecting p-adic and quantum group theories.

Findings

Derived geometric Casselman-Shalika type results for metaplectic covers.

Proved a variant of G. Savin's local Shimura correspondence at the Whittaker level.

Established a Kazhdan-Lusztig theory with new generic parameters for these modules.

Abstract

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 and spherical level. We express our answer in terms of the representation theory of a 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 and develop 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 representation theory of quantum groups. As an application of our results, we deduce geometric Casselman-Shalika type results for metaplectic covers, conjectured in a slightly different form by S. Lysenko, as well as prove a variant of G. Savin's local Shimura type correspondences at the Whittaker level.