Testing rationality of coherent cohomology of Shimura varieties

TL;DR

The paper proves a new rationality criterion for coherent cohomological automorphic forms on Shimura varieties by analyzing pairings between automorphic representations of related groups using ergodic and positivity methods.

Contribution

It introduces a novel approach combining ergodic theory, positivity, and classification of discrete series to establish rationality criteria for automorphic forms on Shimura varieties.

Findings

Existence of non-trivial pairings between automorphic representations of G and G'

Explicit rationality criteria for coherent cohomological forms

Application to inner forms of unitary groups

Abstract

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 $G$.