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

Valentin Buciumas; Manish M. Patnaik

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arXiv · 2211.03724

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

Valentin Buciumas·Manish M. Patnaik

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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.

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BibTeX RIS

Valentin Buciumas, Manish M. Patnaik. 2022-11-07. Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity. https://arxiv.org/abs/2211.03724

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