Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$
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| Format: | Preprint |
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2024
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| _version_ | 1866916538937769984 |
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| author | Balasubramanian, Kumar Pandey, Sanjeev Kumar Joshi, Renu Vasudevan, Varsha |
| author_facet | Balasubramanian, Kumar Pandey, Sanjeev Kumar Joshi, Renu Vasudevan, Varsha |
| contents | Let $F$ be a non-Archimedean local field of characteristic zero and $G=\GL(2,F)$. Let $n\geq 2$ be a positive integer and $\widetilde{G}=\widetilde{\GL}(2,F)$ be the $n$-fold metaplectic cover of $G$. Let $π$ be an irreducible smooth representation of $G$ and $π^{\vee}$ be the contragredient of $π$. Let $τ$ be an involutive anti-automorphism of $G$ satisfying $π^τ\simeq π^{\vee}$. In this case, we say that $τ$ is a dualizing involution. A well known theorem of Gelfand and Kazhdan says that the standard involution $τ$ on $G$ is a dualizing involution. In this paper, we show that any lift of the standard involution to $\widetilde{G}$ is a dualizing involution if and only if $n=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$ Balasubramanian, Kumar Pandey, Sanjeev Kumar Joshi, Renu Vasudevan, Varsha Representation Theory 22E50 Let $F$ be a non-Archimedean local field of characteristic zero and $G=\GL(2,F)$. Let $n\geq 2$ be a positive integer and $\widetilde{G}=\widetilde{\GL}(2,F)$ be the $n$-fold metaplectic cover of $G$. Let $π$ be an irreducible smooth representation of $G$ and $π^{\vee}$ be the contragredient of $π$. Let $τ$ be an involutive anti-automorphism of $G$ satisfying $π^τ\simeq π^{\vee}$. In this case, we say that $τ$ is a dualizing involution. A well known theorem of Gelfand and Kazhdan says that the standard involution $τ$ on $G$ is a dualizing involution. In this paper, we show that any lift of the standard involution to $\widetilde{G}$ is a dualizing involution if and only if $n=2$. |
| title | Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$ |
| topic | Representation Theory 22E50 |
| url | https://arxiv.org/abs/2412.17311 |