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

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 $S{p}_{4n}(\mathbb{A})$ through the application of the theta correspondence and Howe duality to the reductive dual pair $(O_{2n}(\mathbb{A}),S{p}_{4n}(\mathbb{A}))$. The construction involves constructing a non-trivial automorphic character of $O_{2n}(\mathbb{A})$, lifting it to an irreducible cuspidal automorphic representation $\pi ={\otimes }_{\nu }^{\mathrm{\prime }}{\pi }_{\nu }$ of $S{p}_{4n}(\mathbb{A})$, and providing a detailed characterization of the representations ${\pi }_{\nu }$ of $S{p}_{4n}({\mathbb{F}}_{\nu })$ at almost all places $\nu$.