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

Olof Ahlén, Henrik P. A. Gustafsson, Axel Kleinschmidt, Baiying Liu, Daniel Persson

Published 2017-07-27Version 1

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.