Petersson Inner Products and Whittaker--Fourier Periods on Even Special Orthogonal and Symplectic Groups

In this article, we would like to formulate a relation between the square norm of Whittaker–Fourier coefficients on even special orthogonal and symplectic groups and Petersson inner products along with the critical value of $L$-functions up to constants. We follow the path of Lapid and Mao to reduce it to the conjectural local identity. Our strategy is based on the work of Ginzburg–Rallis–Soudry on automorphic descent. We present the analogue result for odd special orthogonal groups, which is conditional on unfolding Whittaker functions of descents.