Abstract: 2d mirror symmetry relates the algebraic and symplectic geometry of a pair of Kähler manifolds obtained by dualizing a Lagrangian torus fibration. We will discuss some known and expected results in the case when the spaces involved are actually hyperkähler, arising as moduli spaces of 4-dimensional supersymmetric gauge theories (for instance, by a K-theoretic version of the Braverman-Finkelberg-Nakajima construction), and the torus fibration is a Hitchin system. In this case, relation between the dual spaces can be seen as a Langlands duality, and many structures of interest in representation theory appear.