On Fourier Coefficients of Automorphic Forms of GL(n)

It is a well-known theorem, due to J. Shalika and I. Piatetski-Shapiro, independently, that any nonzero cuspidal automorphic form on ${\rm GL}_n(\mathbb A)$ is generic, that is, has a nonzero Whittaker–Fourier coefficient. Its proof follows from the Fourier expansion of the cuspidal automorphic form in terms of its Whittaker–Fourier coefficients. In this paper, we extend this Fourier expansion to the whole discrete spectrum of the space of all square-integrable automorphic forms of ${\rm GL}_n(\mathbb A)$ and determine the Fourier coefficients of irreducible noncuspidal (residual) automorphic representations of ${\rm GL}_n(\mathbb A)$ in terms of unipotent orbits.