A general framework and examples of the analytic Langlands correspondence

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.