Convoluted Fourier Coefficients of GL(n)-Automorphic Functions. Part 1

TL;DR

This paper investigates specific convoluted Fourier coefficients of GL(n)-automorphic functions, establishing identities relating them to unipotent orbit Fourier coefficients, with some results supported by conjectures and special cases.

Contribution

It introduces new identities connecting convoluted Fourier coefficients to unipotent orbit coefficients for GL(n), including the most general case studied and some verified special cases.

Findings

Identified identities for convoluted Fourier coefficients in terms of unipotent orbit coefficients.

Established results for the case (n)∘(k,2^{n-1}) up to a stated conjecture.

Provided examples and special cases with proven results.

Abstract

We study certain cases of convoluted Fourier coefficients of $GL_n$-automorphic functions. We establish identities that express them in terms of Fourier coefficients related to unipotent orbits. The most general case that is studied is $(n)\circ(k,2^{n-1})$. The conclusions for this case is only up to a conjecture that I state. However there are certain special cases and other examples that are not based on any conjecture.