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In this paper, we give a proof of the Gan-Gross-Prasad conjecture for thediscrete series of U(p,q). Given a discrete series representation $D(\lambda)$in terms of the Harish-Chandra parameter, the restriction of $D(\lambda)$ toU(p-1,q) contains $D(\mu)$ as a subrepresentation if and only if $\lambda$ and$\mu$ interlaces in a very special way.

Gan-Gross-Prasad Conjecture for U(p,q)