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