On Fourier Coefficients of GL(n)-Automorphic Functions over Number Fields

Eleftherios Tsiokos

Published 2017-11-30Version 1

We study Fourier coefficients of $GL_n(\A)$-automorphic functions $\phi$, for $\A$ being the adele group of a number field $\kkk$. Let FC be an abbreviation for such a Fourier coefficient (and FCs for plural). Roughly speaking, in the present paper we process FCs by iteratively using the operations: Fourier expansions, certain exchanges of Fourier expansions, and conjugations. In Theorem \ref{general} we express any FC in terms of---degenerate in many cases---Whittaker FCs. For FCs obtained from the trivial FC by choosing a certain "generic" term in each Fourier expansion involved, we establish a shortcut (Main corollary \ref{maincor}) for studying their expressions of the form in Theorem \ref{general}. The shortcut gives considerably less information, but it remains useful on finding automorphic representations so that for appropriate choices of $\phi$ in them, the FC is factorizable and nonzero. Then in Theorems \ref{thD1}, \ref{thD2}, and \ref{thD3}, we study examples of FCs on which this shortcut applies, with many of them turning out to "correspond" to more than one unipotent orbit in $GL_n.$ For most of the paper, no knowledge on automorphic forms is necessary.