  In this paper, we construct the ADHM quiver representations and the
corresponding sheaves as the mirror objects of formal deformations of the
framed immersed Lagrangian sphere decorated with flat bundles. More generally,
framed double quivers of Nakajima are constructed as localized mirrors of
framed Lagrangian immersions in dimension two. This produces a localized mirror
functor to the dg category of modules over the framed preprojective algebra.
  For affine ADE quivers in specific multiplicities, the corresponding
(unframed) Lagrangian immersions are homological tori, whose moduli of stable
deformations are asymptotically locally Euclidean (ALE) spaces. We show that
framed stable Lagrangian branes are transformed into monadic complexes of
framed torsion-free sheaves over the ALE spaces.
  A main ingredient is the notion of framed Lagrangian immersions. Moreover, it
is important to note that the deformation space of a Lagrangian immersion with
more than one component is stacky. Using the formalism of quiver algebroid
stacks, we find isomorphisms between the moduli of stable Lagrangian immersions
and that of special Lagrangian fibers of an SYZ fibration in the affine $A_n$
cases.
