There are multiple facets of mirror symmetry: relations between Hodge diamonds, dual Lagrangian fibrations of Strominger-Yau-Zaslow, homological mirror symmetry of Kontsevich. These facets manifest themselves in particular ways for hyperkahler varieties. In this talk I will focus on mirror symmetry relations between dual (holomorphic) Lagrangian fibrations.
For example, the generalized Kummer variety K_n of an abelian surface A is the fibre of the natural map Hilb^{n+1}A->Sym^{n+1}A->A. Debarre described a Lagrangian fibration on K_n whose dual can be constructed in a similar way to the duality between SL- and PGL-Hitchin systems. One can also find isotrivial Lagrangian fibrations on K_n, and their dual fibrations. We verify in numerous cases that these dual fibrations are mirror symmetric, in the sense that their (stringy) Hodge numbers are equal.