  Let $G' \subset G$ be an inclusion of reductive groups whose real points have
a non-trivial discrete series. Combining ergodic methods of Burger-Sarnak and
the author with a positivity argument due to Li and the classification of
minimal $K$-types of discrete series, due to Salamanca-Riba, we show that, if
$\pi$ is a cuspidal automorphic representation of $G$ whose archimedean
component is a sufficiently general discrete series, then there is a cuspidal
automorphic representation of $G'$, of (explicitly determined) discrete series
type at infinity, that pairs non-trivially with $\pi$. When $G$ and $G'$ are
inner forms of U(n) and $U(n-1)$, respectively, this result is used to define
rationality criteria for sufficiently general coherent cohomological forms on
$G$.
