On the length of global integrals for $GL_n$

TL;DR

This paper proves a conjecture about the length of global integrals for certain representations of the group $GL_n$, advancing understanding in automorphic forms and representation theory.

Contribution

It establishes the conjecture for specific cases, including when the integral length is four, and discusses the general case for $GL_n$ representations.

Findings

Proved the conjecture for integrals of length four.

Extended the proof to various cases in the general setting.

Provided insights into the structure of global integrals for $GL_n$.

Abstract

In this paper we prove Conjecture \ref{conj1} for a set of representations of the group $GL_n({\bf A})$. This Conjecture is stated in complete generality as Conjecture 1 in \cite{G2}, and here we prove it for various cases. See Conjecture \ref{conj2} below. First we prove it in the case when the length of the integral is four, and then we discuss the general case.