Beilinson–Bernstein Localization Over ℚ and Periods of Automorphic Forms

This article explains an approach to Beilinson–Bernstein localization over $\mathbb Q$, when G is the group attached to a Shimura variety S(G,X). We develop a theory of compatible rational structures, over appropriately chosen number fields, on discrete series Harish-Chandra modules over the groups attached to the conjugate Shimura varieties. In this way, we can endow automorphic representations of discrete series type at infinity with canonical rational structures over natural coefficient fields, and show that these are compatible with Galois conjugation.