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

TL;DR

This paper establishes a relation between Betti-Whittaker periods for automorphic representations of GL_n and their duals, leading to new insights into the algebraicity of critical L-value ratios.

Contribution

It proves a period relation under duality for Betti-Whittaker periods of GL_n automorphic representations, with implications for L-value algebraicity.

Findings

Proves a period relation between Betti-Whittaker periods and their contragredients.

Shows trivialness of a certain relative period for GL_{2n} of orthogonal type.

Demonstrates algebraicity of ratios of successive critical L-values.

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.