  In this paper, we investigate when weighted composition operators acting on
Dirichlet spaces $\mathcal{D}(\mathbb{B}_{N})$ are complex symmetric with
respect to some special conjugations, and provide some characterizations of
Hermitian weighted composition operators on $\mathcal{D}(\mathbb{B}_{N})$.
Furthermore, we give a sufficient and necessary condition for $J$-symmetric
weighted composition operators on Hardy spaces $H^2(\mathbb{B}_{N})$ to be
unitary or Hermitian, then some new examples of complex symmetric weighted
composition operators on $H^2(\mathbb{B}_{N})$ are obtained. We also discuss
the normality of complex symmetric weighted composition operators on
$H^2(\mathbb{B}_{N})$.
