  We examine the CP properties of chiral gauge theory defined by a formulation
of the domain wall fermion, where the light field variables $q$ and $\bar q$
together with Pauli-Villars fields $Q$ and $\bar Q$ are utilized. It is shown
that this domain wall representation in the infinite flavor limit $N=\infty$ is
valid only in the topologically trivial sector, and that the conflict among
lattice chiral symmetry, strict locality and CP symmetry still persists for
finite lattice spacing $a$. The CP transformation generally sends one
representation of lattice chiral gauge theory into another representation of
lattice chiral gauge theory, resulting in the inevitable change of propagators.
A modified form of lattice CP transformation motivated by the domain wall
fermion, which keeps the chiral action in terms of the Ginsparg-Wilson fermion
invariant, is analyzed in detail; this provides an alternative way to
understand the breaking of CP symmetry at least in the topologically trivial
sector. We note that the conflict with CP symmetry could be regarded as a
topological obstruction. We also discuss the issues related to the definition
of Majorana fermions in connection with the supersymmetric Wess-Zumino model on
the lattice.
