Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups

This work is largely inspired by the 2003 Ph.D. thesis \cite{snitz} of Kobi Snitz. In his thesis, Snitz constructed two irreducible, automorphic, cuspidal representations $\pi$ and ${\pi }^{\mathrm{\prime }}$ of the metaplectic group $G\left(\mathbb{A}\right)={\widetilde{SL}}_2\left(\mathbb{A}\right)$ where each representation is obtained from a different global theta lifts of certain non-trivial automorphic characters $\xi$ and ${\xi }^{\mathrm{\prime }}$ of the orthogonal groups $H_{\mathbb{A}}=O(q,V)\left(\mathbb{A}\right)$ and $H_{\mathbb{A}}^{\mathrm{\prime }}=O(q^{\mathrm{\prime }},V^{\mathrm{\prime }})\left(\mathbb{A}\right)$, respectively, where $\mathbb{A}={\mathbb{A}}_{\mathbb{F}}$ is the adele ring of a number field $\mathbb{F}$. Snitz shows that for certain matching data of quadratic spaces and automorphic quadratic characters, that these two representations of $G\left(\mathbb{A}\right)$ are isomorphic, i.e. $\pi \cong {\pi }^{\mathrm{\prime }}$. The goal of this work is to reformulate and generalize Snitz’s work to higher rank groups. Namely we wish to determine for which admissible data $((q,V),\xi ,(q^{\mathrm{\prime }},V^{\mathrm{\prime }}),{\xi }^{\mathrm{\prime }})$ satisfying certain local necessary conditions could an isomorphism possibly exist between two global theta lifts $\pi$ and ${\pi }^{\mathrm{\prime }}$ with respect to two reductive dual pairs $H_{\mathbb{A}}\times G_{\mathbb{A}}$ and $H_{\mathbb{A}}^{\mathrm{\prime }}\times G_{\mathbb{A}}$ and two non-trivial automorphic quadratic characters $\xi$ and ${\xi }^{\mathrm{\prime }}$ of the orthogonal groups $H_{\mathbb{A}}=O(q,V)\left(\mathbb{A}\right)$ and $H_{\mathbb{A}}^{\mathrm{\prime }}=O(q^{\mathrm{\prime }},V^{\mathrm{\prime }})\left(\mathbb{A}\right)$, respectively and the group $G$ which is the symplectic or the metaplectic group.