Testing rationality of coherent cohomology of Shimura varieties

Let $G^{\mathrm{\prime }}\subset G$ 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 $G^{\mathrm{\prime }}$, of (explicitly determined) discrete series type at infinity, that pairs non-trivially with $\pi$. When $G$ and $G^{\mathrm{\prime }}$ 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 $G$.