Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality

Abstract: 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 ${\rm GL}n(\mathbb{A})$ and its contragredient. As a consequence, we obtain the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for ${\rm GSpin}{2n}^* \times {\rm GL}{n'}$ by the result of Harder and Raghuram.