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Compatibility of theta lifts and tempered condition

Part of: Lie groups

Published online by Cambridge University Press:  21 June 2023

Zhe Li  and
Shanwen Wang Open the ORCID record for Shanwen Wang [Opens in a new window]
Zhe Li
Affiliation:
School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, China e-mail: zli17@fudan.edu.cn
Shanwen Wang*
Affiliation:
School of Mathematics, Renmin University of China, No. 59, Zhongguancun Street, Haidian District, Beijing 100872, China
*
e-mail: s_wang@ruc.edu.cn
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Abstract

In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, we show that, for metapletic–orthogonal dual pair over $\mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

Keywords

Tempered representationstheta correspondencemetaplectic groupsWeil representations

MSC classification

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 60 - 73
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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