I am wondering if there is some relationship between Serre duality and Pontragin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator with the Laplace operator. While differential operators can be written as pseudo-differential operators with symbols in the cotangent bundle, I do not know how this is related to the usual character rhetoric characterization of Pontragin duality. 

A compact manifold is usually not a group, and there is no analgous Haar measure on it. Since Fourier inversion still holds for a suitable class of functions, I am wondering if there is any way to extend the Pontragin duality to more "global" objects. Is it possible to incorporate the notion of cotangent bundle (or dualizing sheaf) in Pontragin duality? Is it possible to interpret symbol of operators corresponding to elements in some "dual group"? I am not sure if this is a well-investigated topic, however after extensive literature search I did not find anything. 

I am totally okay with a pure algebraic perspective without using analytical part of Hodge theory (say using Gronthendieck duality or something).