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

References in corpus (2)

• On Deligne's periods for tensor product motives
• On the intersection of local Arthur packets for classical groups and applications