Analytic Langlands Correspondence for G=PGL(2,C)

Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan based on ideas and results of Langlands, Teschner, Braverman-Kazhdan and Kontsevich. Let X be a smooth irreducible projective curve over C, G be a semisimple group. On one side of this conjectural correspondence there are G^-opers on X satisfying a certain condition (real opers), where G^ is Langlands dual group. On the other side there are certain operators on L^2(Bun_G), called Hecke operators, where Bun_G is the variety of stable G-bundles on X and L^2(Bun_G) is a Hilbert space of square-integrable half-densities. I will describe the main picture and present new results in this direction. Partially based on joint projects with A. Wang and S. Raman.