  In this paper, we define the induced modules of Lie algebra ad$(B)$
associated with a 3-Lie algebra $B$-module, and study the relation between
3-Lie algebra $A_{\omega}^{\delta}$-modules and induced modules of inner
derivation algebra ad$(A_{\omega}^{\delta})$. We construct two infinite
dimensional intermediate series modules of 3-Lie algebra $A_{\omega}^{\delta}$,
and two infinite dimensional modules $(V, \psi_{\lambda\mu})$ and $(V,
\phi_{\mu})$ of the Lie algebra
  ad$(A_{\omega}^{\delta})$, and prove that only $(V, \psi_{\lambda0})$ and
$(V, \psi_{\lambda1})$ are induced modules.
