Analytic Langlands correspondence for $$PGL_2$$ P G L 2 on $${\mathbb {P}}^1$$ P 1 with parabolic structures over local fields

Abstract

We continue to develop the analytic Langlands program for curves over local fields initiated in our earlier papers, following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators which we introduced in those papers in the case of a projective line with parabolic structures at finitely many points for the group \(PGL_2\). We establish most of our conjectures in this case.