Langlands duality for Hitchin systems

Consider a complex smooth projective curve X and the stack Bun_n(X) of
rank n vector bundles on X. Let T*Bun_n(X) be the cotangent stack of X
(this is the phase space of the Hitchin system). Then the Langlands
duality for Hitchin systems is an auto-equivalence of the derived
category of T*Bun_n(X) satisfying certain properties.

The Langlands duality for Hitchin systems is the classical limit of
the famous Langlands duality. I will explain the formulation in
detail. Then we will switch to a (mostly conjectural) local duality,
where X is replaced by a formal disc.

I will assume only very basic knowledge of algebraic geometry.