The classical limit of the global geometric Langlands correspondence is a conjectural Fourier-Mukai equivalence between the Hitchin fibrations for a reductive group G and its dual. This "global" statement can sometimes be approached by global tools, but it is also expected to be compatible with the "local" claim (concerning moduli of Higgs bundles on formal disk).

In my talk, I will describe the current status of the conjecture.