  Piatetski-Shapiro the concept of CAP representations was introduced,
elucidating the Saito-Kurokawa representations of $PGSp(4)$. In this paper we
present a family of CAP representations for the group $Sp_{4n}(\mathbb A)$
through the application of the theta correspondence and Howe duality to the
reductive dual pair $(O_{2n}(\mathbb A) , Sp_{4n}(\mathbb A))$. The
construction involves constructing a non-trivial automorphic character of
$O_{2n}(\mathbb A)$, lifting it to an irreducible cuspidal automorphic
representation $\pi=\otimes'_{\nu}\pi_{\nu}$ of $Sp_{4n}(\mathbb A)$, and
providing a detailed characterization of the representations $\pi_{\nu}$ of
$Sp_{4n}(\mathbb F_\nu)$ at almost all places $\nu$.
