We consider the problem of minimal Lagrangian immersions of disks into CH^2 which are equivariant to some surface group representation. We prove several results on existence and (non)uniqueness. The local parameterization of the immersion is given by the conformal structure on a closed surface and a holomorphic cubic differential on that conformal structure, hence of complex dimension 8g-8, where g>1 is the genus. This is a joint work with John Loftin and Marcello Lucia.