On a refined local converse theorem for SO(4)
Abstract
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 $\pi$ 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 $\pi$.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2023
- DOI:
- 10.48550/arXiv.2302.06256
- arXiv:
- arXiv:2302.06256
- Bibcode:
- 2023arXiv230206256Y
- Keywords:
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- Mathematics - Representation Theory
- E-Print:
- Conjecture 1.2 in a previous draft was removed since we were told that it is false even for SO(6)