  We study complex Lagrangian submanifolds of a compact hyper-K\"ahler manifold
and prove two results: (a) that an involution of a hyper-K\"ahler manifold
which is antiholomorphic with respect to one complex structure and which acts
non-trivially on the corresponding symplectic form always has a fixed point
locus which is complex Lagrangian with respect to one of the other complex
structures, and (b) there exist Lagrangian submanifolds which are complex with
respect to one complex structure and are not the fixed point locus of any
involution which is anti-holomorphic with respect to one of the other complex
structures.
