Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity
Valentin Buciumas, Manish M. Patnaik

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 -fold metaplectic cover of a -adic group 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 . 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 -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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
