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

An Explicit Construction of CAP Representations of $Sp_{4n}(\mathbb A)$  associated to Non-trivial Automorphic Characters of Orthogonal Groups  $O_{2n}(\mathbb A)$