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In this paper, we prove one direction of the Gan--Gross--Prasad conjecture onmetaplectic-symplectic groups for tempered cases. As an application, we discussthe non-vanishing of the central value of quadratic twists of automorphic$L$-functions of $GL_{2n}$.

The Fourier-Jacobi Periods : The case of $Mp(2n+2r) \times Sp(2n)$