On the genericity of Eisenstein series and their residues for covers of GL_m

Let τ₁^(r), τ₂^(r) be two genuine cuspidal automorphic representations on r-fold covers of the adelic points of the general linear groups GL_n1, GL_n2, respectively, and let E(g, s) be the associated Eisenstein series on an r-fold cover of GL_n1+n2, normalized to have functional equation under s ⟼ 1 - s. Suppose R(s) ≥ 1/2 without loss. Then the value at any point of holomorphy s = s₀ or the reside at any (simple) pole of E(g, s) is an automorphic form, and generates an automorphic representation. In this note we show that if n₁ ≠ n₂ these automorphic representations (when not identically zero) are generic, while if n₁ = n₂ := n they are generic except for residues at s = n+1/2n.