$L$-functions and periods of adjoint motives

Abstract

The article studies the compatibility of the refined Gross–Prasad (or Ichino–Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of $L$-functions. When the automorphic representations are of motivic type, it is shown that the $L$-values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) $L$-functions.