| 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 pi of general linear groups or classical groups, characterizing the set nm(pi) 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.;In this thesis, first, we extend the Fourier expansion of cuspidal automorphic forms of GLn( A) to any automorphic form occurring in the discrete spectrum of GLn( A).;Then, we determine the set nm(pi) for any residual representation Delta(tau, m) of GL2mn( A) (with tau&;Next, we consider certain set ofirreducible cuspidal automorphic representations of Sp 4mn( A) which are nearly equivalent to the residual representation EDt,m . We show that this set decomposes naturally into two disjoint sets, corresponding to certain sets of irreducible cuspidal automorphic representations of Sp&d15;4mn-2n A and Sp&d15;4mn+2n A, respectively. This extends the Ginzburg-Jiang-Soudry correspondences between certain automorphic forms on Sp4 n( A) and Sp&d15;2n A.;At last, we recall Arthur's classification of the discrete spectrum and a conjecture of Jiang towards understanding Fourier coefficients of automorphic forms in automorphic L2-packets, and briefly discuss the relation between them and the above results in this thesis. |
