On the Genericity of Eisenstein Series and Their Residues for Covers of $GL_m$

TL;DR

This paper investigates the genericity of automorphic representations generated by Eisenstein series and their residues on covers of $GL_m$, revealing conditions under which these representations are generic.

Contribution

It demonstrates that automorphic representations from Eisenstein series are generally generic, with specific exceptions for residues when the ranks are equal.

Findings

Automorphic representations are generic when $n_1 eq n_2$.

Representations are mostly generic for $n_1 = n_2$, except at certain residues.

Residues at specific points $s=rac{n eq 1}{2n}$ are non-generic.

Abstract

Let $\tau_1^{(r)}$, $\tau_2^{(r)}$ be two genuine cuspidal automorphic representations on $r$-fold covers of the adelic points of the general linear groups $GL_{n_1}$, $GL_{n_2}$, resp., and let $E(g,s)$ be the associated Eisenstein series on an $r$-fold cover of $GL_{n_1+n_2}$. Then the value or residue at any point $s=s_0$ of $E(g,s)$ is an automorphic form, and generates an automorphic representation. In this note we show that if $n_1\neq n_2$ these automorphic representations (when not identically zero) are generic, while if $n_1=n_2:=n$ they are generic except for residues at $s=\frac{n\pm1}{2n}$.