Algebraicity of adjoint $L$-functions for quasi-split groups

Abstract

For a globally generic cuspidal automorphic representation $\mathit{\Pi}$ of a quasi-split reductive group $G$ over $\mathbb Q$, E. Lapid and Z. Mao proposed a conjecture on the decomposition of the global Whittaker functionals on $\mathit{\Pi}$ into products of an adjoint $L$-value of $\mathit{\Pi}$ and the local Whittaker functionals. In this paper, we consider the algebraic aspect of the Lapid-Mao conjecture. More precisely, when $\mathit{\Pi}$ is $C$-algebraic, we show that the algebraicity of the adjoint $L$-value can be expressed in terms of the Petersson norm of Whittaker-rational cusp forms in $\mathit{\Pi}$, subject to the validity of the Lapid-Mao conjecture. For unitary similitude groups, we also establish an unconditional and more refined algebraicity result. Additionally, we give an explicit formula for the case $G={\rm U}(2,1)$.