Period Relations Between the Betti–Whittaker Periods for GLn Under Duality

In this paper, under some regularity conditions, we prove a period relation between the Betti–Whittaker periods associated to a regular algebraic cuspidal automorphic representation of GLₙ(A) and its contragredient. As a consequence, we prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of GL₂ₙ(A) of orthogonal type, which implies the algebraicity of the ratios of successive critical L-values for GSpin₂ₙ* × GL_n^ by the result of Harder and Raghuram.