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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.13469 |
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| _version_ | 1866913207542611968 |
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| author | Erez, Ron |
| author_facet | Erez, Ron |
| contents | 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 $ π$ and $ π' $ 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 $ ξ$ and $ ξ' $ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V' \right ) \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. $ π\congπ'$. 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 $\left ( \left ( q, V \right ) ,ξ,
\left ( q', V' \right ),ξ'\right )$ satisfying certain local necessary conditions could an isomorphism possibly exist between two global theta lifts $ π$ and $ π'$ with respect to two reductive dual pairs $ H_{ \mathbb A } \times G_{ \mathbb A } $ and $ H'_{ \mathbb A } \times G_{ \mathbb A } $ and two non-trivial automorphic quadratic characters $ ξ$ and $ ξ'$ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V'\right ) \left ( \mathbb A \right ) $, respectively and the group $ G $ which is the symplectic or the metaplectic group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13469 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups Erez, Ron Representation Theory Number Theory 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 $ π$ and $ π' $ 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 $ ξ$ and $ ξ' $ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V' \right ) \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. $ π\congπ'$. 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 $\left ( \left ( q, V \right ) ,ξ, \left ( q', V' \right ),ξ'\right )$ satisfying certain local necessary conditions could an isomorphism possibly exist between two global theta lifts $ π$ and $ π'$ with respect to two reductive dual pairs $ H_{ \mathbb A } \times G_{ \mathbb A } $ and $ H'_{ \mathbb A } \times G_{ \mathbb A } $ and two non-trivial automorphic quadratic characters $ ξ$ and $ ξ'$ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V'\right ) \left ( \mathbb A \right ) $, respectively and the group $ G $ which is the symplectic or the metaplectic group. |
| title | Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2401.13469 |