Fourier coefficients attached to small automorphic representations of ${mathrm{SL}}_n(mathbb{A})$

We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${mathrm{SL}}_n(mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.