Testing rationality of coherent cohomology of Shimura varieties

Let G' subset of 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.