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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.06256 |
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| _version_ | 1866914308554752000 |
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| author | Yan, Pan Zhang, Qing |
| author_facet | Yan, Pan Zhang, Qing |
| contents | Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for $\mathrm{SO}_{2n}(F)$ over a $p$-adic field $F$, which says that, up to an outer automorphism of $\mathrm{SO}_{2n}(F)$, an irreducible generic representation of $\mathrm{SO}_{2n}(F)$ is uniquely determined by its twisted gamma factors by generic representations of $\mathrm{GL}_k(F)$ for $k=1,\dots,n$. It is desirable to remove the ``up to an outer automorphism" part in the above theorem using more twisted gamma factors, but this seems a hard problem. In this paper, we provide a solution to this problem for the group $\mathrm{SO}_4(F)$, namely, we show that a generic supercuspidal representation $π$ of $\mathrm{SO}_4(F)$ is uniquely determined by its $\mathrm{GL}_1$, $\mathrm{GL}_2$ twisted local gamma factors and a twisted exterior square local gamma factor of $π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06256 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a refined local converse theorem for SO(4) Yan, Pan Zhang, Qing Representation Theory Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for $\mathrm{SO}_{2n}(F)$ over a $p$-adic field $F$, which says that, up to an outer automorphism of $\mathrm{SO}_{2n}(F)$, an irreducible generic representation of $\mathrm{SO}_{2n}(F)$ is uniquely determined by its twisted gamma factors by generic representations of $\mathrm{GL}_k(F)$ for $k=1,\dots,n$. It is desirable to remove the ``up to an outer automorphism" part in the above theorem using more twisted gamma factors, but this seems a hard problem. In this paper, we provide a solution to this problem for the group $\mathrm{SO}_4(F)$, namely, we show that a generic supercuspidal representation $π$ of $\mathrm{SO}_4(F)$ is uniquely determined by its $\mathrm{GL}_1$, $\mathrm{GL}_2$ twisted local gamma factors and a twisted exterior square local gamma factor of $π$. |
| title | On a refined local converse theorem for SO(4) |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2302.06256 |