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

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 $s=\frac{n\pm1}{2n}$.