{"title":"A general framework and examples of the analytic Langlands correspondence","authors":"Pavel Etingof, Edward Frenkel, David Kazhdan","doi":"10.4310/pamq.2024.v20.n1.a8","DOIUrl":null,"url":null,"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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n1.a8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.