  We discuss the perturbative $O(\alpha_s a)$ matching in the static heavy and
domain-wall light quark system. The gluon action is the Iwasaki action and the
link smearing is performed in the static heavy action. The chiral symmetry of
the light quark realized by using the domain-wall fermion formulation does not
prohibit the mixing of the operators at $O(a)$. The application of $O(a)$
improvement to the actual data shows that the B meson decay constant $f_B$, the
matrix elements ${\cal M}_B$ and the B parameter $B_B$ have non-negligible
effects, while the effect on the SU(3) breaking ratio $\xi$ is small.
