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

TL;DR

This paper analyzes Fourier coefficients of automorphic functions on GL(n) over number fields, expressing them in terms of degenerate Whittaker coefficients and providing methods to identify factorizable nonzero coefficients for automorphic representations.

Contribution

It introduces a systematic process to express Fourier coefficients in terms of degenerate Whittaker coefficients and offers a shortcut for identifying factorizable nonzero coefficients in automorphic representations.

Findings

Any Fourier coefficient can be expressed via degenerate Whittaker coefficients.

A shortcut helps find automorphic representations with factorizable nonzero Fourier coefficients.

Many Fourier coefficients correspond to multiple unipotent orbits in GL(n).

Abstract

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 3.1 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 6.17) for studying their expressions of the form in Theorem 3.1. 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 8.3.11, 8.3.12, and 8.3.18, 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.