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We continue to develop the analytic Langlands program for curves over localfields initiated in arXiv:1908.09677, arXiv:2103.01509 following a suggestionof Langlands and a work of Teschner. Namely, we study the Hecke operatorsintroduced in arXiv:2103.01509 in the case of P^1 over a local field withparabolic structures at finitely many points for the group PGL(2). We establishmost of the conjectures of arXiv:1908.09677, arXiv:2103.01509 in this case.

Analytic Langlands correspondence for PGL(2) on P^1 with parabolic structures over local fields