Period relations between the Betti-Whittaker periods for ${\mathrm{G}\mathrm{L}}_n$ 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 ${\mathrm{G}\mathrm{L}}_n(\mathbb{A})$ and its contragredient. As a consequence, we obtain the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of ${\mathrm{G}\mathrm{L}}_{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for ${\mathrm{G}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_{2n}^{\ast }\times {\mathrm{G}\mathrm{L}}_{n^{\mathrm{\prime }}}$ by the result of Harder and Raghuram.