It is a well-known theorem, due to J. Shalika and I. Piatetski-Shapiro, independently, that any nonzero cuspidal automorphic form on 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 and determine the Fourier coefficients of irreducible noncuspidal (residual) automorphic representations of in terms of unipotent orbits.
Bibliographical noteFunding Information:
The work of the first named author is supported in part by NSF DMS-1001672.