A general framework for the analytic Langlands correspondence

We discuss a general framework for the analytic Lang-lands correspondence over an arbitrary local field F introduced and studied in our works [EFK1, EFK2, EFK3], in particular including non-split and twisted settings. Then we specialize to the archimedean cases (F = C and F = 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 [EFK2, EFK3] and [GW]. We also describe an analogue of the Langlands functoriality principle in the analytic Langlands correspondence over C and show that it is compatible with the results and conjectures of [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.