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Published online by Cambridge University Press: 21 June 2023
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.