Betti-Whittaker periods under duality: variations and applications

One of the authors (Chen) had previously proved a result on the behavior of Betti-Whittaker periods under duality for cohomological cuspidal automorphic representations of ${\rm GL}_n/{\mathbb Q}$ under some regularity assumptions while using their relation to $L$-values as an anchor in his proof. In this article we prove a generalization of this result to ${\rm GL}_n$ over any number field $F$ without any regularity assumptions and without recourse to $L$-values, while using the outer-automorphism of ${\rm GL}_n$ as the main tool. Then, using results of Harder and one of the other authors (Raghuram), we give applications to new rationality results for the ratios of special values of general triple product $L$-functions and for general twisted Asai $L$-functions. We also give a new proof of a previous result of Bhagwat and Raghuram on the special values of $L$-functions for orthogonal groups. We present variations on period relations for the Betti-Shalika periods under duality, and the behavior of Betti-Whittaker periods under Galois automorphisms of $F$.