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

TL;DR

This paper demonstrates that Fourier coefficients of automorphic forms linked to minimal or next-to-minimal automorphic representations of SL_n are fully determined by degenerate Whittaker coefficients, providing explicit formulas and potential applications in string theory.

Contribution

It provides explicit formulas for Fourier coefficients of automorphic forms attached to specific automorphic representations of SL_n, extending understanding of their structure and applications.

Findings

Fourier coefficients are determined by degenerate Whittaker coefficients.

Explicit Fourier expansion formulas analogous to classical formulas.

Expressions derived for Fourier coefficients related to all maximal parabolic subgroups.

Abstract

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.