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We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture forunitary groups, by using local theta correspondence to relate theFourier-Jacobi case with the Bessel case established by Beuzart-Plessis. Toachieve this, we prove two conjectures of D. Prasad on the precise descriptionof the local theta correspondence for (almost) equal rank unitary dual pairs interms of the local Langlands correspondence.

The Gross-Prasad conjecture and local theta correspondence