Fourier coefficients of automorphic forms and arthur classification

Abstract

Fourier coefficients play important roles in the study of both classical modular forms and automorphic forms. For example, it is a well-known theorem of Shalika and Piatetski-Shapiro that cuspidal automorphic forms of GLn(A) are globally generic, that is, have non-degenerate Whittaker-Fourier coefficients, which is proved by taking Fourier expansion. For general connected reductive groups, there is a framework of attaching Fourier coefficients to nilpotent orbits. For general linear groups and classical groups, nilpotent orbits are parametrized by partitions. Given any automorphic representation π of general linear groups or classical groups, characterizing the set n^m(π) of maximal partitions with corresponding nilpotent orbits providing non-vanishing Fourier coefficients is an interesting question, and has applications in automorphic descent and construction of endoscopic lifting.