Mirror symmetry relates the algebraic and symplectic geometry of spaces which are related by dualizing a Lagrangian torus fibration. From the perspective of representation theory this is particularly interesting, with ties to geometric Langlands duality, in cases where the spaces are hyperkähler, and the Lagrangian tori are actually holomorphic Lagrangian. Such spaces, which arise as moduli spaces of four-dimensional field theories, include character varieties, multiplicative quiver varieties, and the "K-theoretic Coulomb branches" of Braverman-Finkelberg-Nakajima. We discuss some known and conjectured results in this situation, emphasizing a "microlocal-sheaf" approach to computing Fukaya categories.