A general framework and examples of the analytic Langlands correspondence

Pub Date : 2024-03-26 DOI:10.4310/pamq.2024.v20.n1.a8
Pavel Etingof, Edward Frenkel, David Kazhdan
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Abstract

We discuss a general framework for the analytic Langlands correspondence over an arbitrary local field F introduced and studied in our works [$\href{http://arxiv.org/abs/1908.09677}{EFK1}$, $\href{http://arxiv.org/abs/2103.01509}{EFK2}$, $\href{http://arxiv.org/abs/2106.05243}{EFK3}$], in particular including non-split and twisted settings. Then we specialize to the archimedean cases ($F = \mathbb{C}$ and $F = \mathbb{R}$) and give a (mostly conjectural) description of the spectrum of the Hecke operators in various cases in terms of opers satisfying suitable reality conditions, as predicted in part in [$\href{http://arxiv.org/abs/2103.01509}{EFK2}$, $\href{http://arxiv.org/abs/2106.05243}{EFK3}$] and [$\href{http://arxiv.org/abs/2107.01732}{GW}$]. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over $\mathbb{C}$ and show that it is compatible with the results and conjectures of [$\href{http://arxiv.org/abs/2103.01509}{EFK2}$]. Finally, we apply the tools of the analytic Langlands correspondence over archimedean fields in genus zero to the Gaudin model and its generalizations, as well as their $q$-deformations.
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分析朗兰兹对应关系的一般框架和示例
我们讨论在我们的著作[$\href{http://arxiv.org/abs/1908.09677}{EFK1}$, $\href{http://arxiv.org/abs/2103.01509}{EFK2}$, $\href{http://arxiv.org/abs/2106.05243}{EFK3}$]中引入和研究的任意局部域 F 上的解析朗兰兹对应关系的一般框架,特别是包括非分裂和扭曲的情形。然后,我们专门讨论了阿基米德情况($F = \mathbb{C}$和$F = \mathbb{R}$),并根据 [$\href{http://arxiv.org/abs/2103.01509}{EFK2}$, $\href{http://arxiv.org/abs/2106.05243}{EFK3}$] 和 [$\href{http://arxiv.org/abs/2107.01732}{GW}$]中的部分预测,以满足合适现实条件的运算符为条件,给出了各种情况下赫克算子谱的描述(主要是猜想)。我们还描述了在 $\mathbb{C}$ 上的解析朗兰兹对应关系中朗兰兹函数性原理的一个类比,并证明它与 [$\href{http://arxiv.org/abs/2103.01509}{EFK2}$] 的结果和猜想是兼容的。最后,我们将零属的拱门域上的解析朗兰兹对应的工具应用于高丁模型及其广义,以及它们的 $q$ 变形。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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