论文标题
$ p $ adadic群体和量子组的元容器封面
Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity
论文作者
论文摘要
我们在Iwahori和球形级别的$ n $ fold Metapic盖上描述了Whittaker或Gelfand-Graev模块的结构。我们以$ g $的兰格兰双重组的统一根部的量子组的表示理论来表达我们的答案。为此，我们引入了这些模块的代数组合模型，并为它们开发了涉及新通用参数的Kazhdan-Lusztig理论。这些参数可以专门用于高斯总和以恢复$ p $ - ad的理论，也可以恢复量子组表示理论中的自然分级参数。作为结果的应用，我们推定了元容器覆盖物的几何Casselman-Shalika类型结果，该结果以S. lysenko的形式略有不同，并证明了G. Savin的局部Shimura类型对应关系的变体。
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.
