On Unfoldings of Some Integrals of Automorphic Functions on General Linear Groups

TL;DR

This paper investigates certain integrals of automorphic functions on general linear groups, utilizing Fourier coefficients to gain new insights into their structure and properties.

Contribution

It introduces new methods for analyzing integrals of automorphic functions using Fourier coefficients, extending previous results and providing deeper understanding of their unfoldings.

Findings

Derived new formulas for integrals involving Fourier coefficients

Extended Fourier coefficient analysis to broader classes of automorphic forms

Provided new insights into the structure of automorphic integrals on GL(n)

Abstract

We use results about Fourier coefficients appearing in [T] (and some more obtained here), to obtain information for certain among the integrals of the form $$I=\int_{GL_n(\kkk)Z_n(\A)\s GL_n(\A)}\varphi(g)\phi(g)\F(E)(\tj(g))dg$$ where: $\A$ is the adele ring of a number field $\kkk$; $\varphi$ is a $GL_n(\A)$-cuspidal automorphic form; $\phi$ is a $GL_n(\A)$-automorphic function (even the trivial for some results); $E$ is a $GL_{N}(\A)$-automorphic form for a multiple $N$ of $n$; $\F(E)$ is a Fourier coefficient of $E$ for certain choices of additive functions $\F$ in a set $\BBnk[N]$ which we defined in $[T];$ $\tj$ is a diagonal embedding of $GL_n$ in $GL_N$; of course $\tj(GL_n)\in\Stab{GL_N}{\F}$; and $Z_n$ is the center of $GL_n$.