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.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/158