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The local converse theorem for quasi-split $\mathrm {O}_{2n}$ and $\mathrm {SO}_{2n}$

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  07 April 2025

Jaeho Haan Open the ORCID record for Jaeho Haan [Opens in a new window] ,
Yeansu Kim Open the ORCID record for Yeansu Kim [Opens in a new window]  and
Sanghoon Kwon Open the ORCID record for Sanghoon Kwon [Opens in a new window]
Jaeho Haan
Affiliation:
Department of Mathematics Education, Catholic Kwandong University, Gangneung, South Korea e-mail: jaehohaan@gmail.com skwon@cku.ac.kr
Yeansu Kim*
Affiliation:
Department of Mathematics Education, Chonnam National University, Gwangju, South Korea
Sanghoon Kwon
Affiliation:
Department of Mathematics Education, Catholic Kwandong University, Gangneung, South Korea e-mail: jaehohaan@gmail.com skwon@cku.ac.kr
*
e-mail: ykim@jnu.ac.kr
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Abstract

Let F be a non-archimedean local field of characteristic not equal to 2. In this article, we prove the local converse theorem for quasi-split $\mathrm {O}_{2n}(F)$ and $\mathrm {SO}_{2n}(F)$, via the description of the local theta correspondence between $\mathrm {O}_{2n}(F)$ and $\mathrm {Sp}_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma $-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\mathrm {O}_{2n}(\mathbb {A})$ and $\mathrm {SO}_{2n}(\mathbb {A})$, respectively, where $\mathbb {A}$ is a ring of adele of a global number field L.

Keywords

Gamma factorseven orthogonal groupslocal converse theoremtheta correspondence

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

J. H. has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01048645). Y. K. has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. RS-2022-0016551 and No. RS-2024-00415601 (G-BRL)) and by Chonnam National University (Grant number: 2022-0123). J. H. and S. K. have been supported by NRF grant (No. RS-2023-00237811).

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