Algebraicity of adjoint L-functions for quasi-split groups

For a globally generic cuspidal automorphic representation Π of a quasi-split reductive group G over Q, E. Lapid and Z. Mao proposed a conjecture on the decomposition of the global Whittaker functionals on Π into products of an adjoint L-value of Π and the local Whittaker functionals. In this paper, we consider the algebraic aspect of the Lapid-Mao conjecture. More precisely, when Π 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 Π, 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= U(2,1).