Testing rationality of coherent cohomology of Shimura varieties

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 $π$ 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 $π$. 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 $G$.

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