A motivic interpretation of Whittaker periods for ${\mathrm{G}\mathrm{L}}_n$

Admitting the existence of conjectural motives attached to cohomological irreducible cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_n$, we write down Raghuram and Shahidi’s Whittaker periods in terms of Yoshida’s fundamental periods when the base field is a totally real number field or a CM field.