Among abelian varieties, Jacobians of smooth projective curves C have the important property of being autodual, i.e., they are canonically isomorphic to their dual abelian varieties. This is equivalent to the existence of a Poincaré line bundle P on J(C)×J(C) which is universal as a family of algebraically trivial line bundles on J(C). A yet other instance of this fact was discovered by S. Mukai, who proved that the Fourier-Mukai transform with kernel P is an auto-equivalence of the bounded derived category of J(C). I will talk on joint work with Filippo Viviani and Antonio Rapagnetta, where we try to generalize both the autoduality result and Mukai’s equivalence result for singular reducible curves X with locally planar singularities. Our results generalize previous work of Arinkin, Esteves, Gagné, Kleiman and can be seen as an instance of the geometric Langlands duality for the Hitchin fibration.