Poles of Eisenstein series on general linear groups induced from two Speh representations

David Ginzburg; David Soudry

paper

Poles of Eisenstein series on general linear groups induced from two Speh representations

arXiv:2410.23026

Abstract

We determine the poles of the Eisenstein series on a general linear group, induced from two Speh representations, $Δ (τ, m_{1}) ∣ \cdot ∣^{s} \times Δ (τ, m_{2}) ∣ \cdot ∣^{- s}$ , $R e (s) \geq 0$ , where $τ$ is an irreducible, unitary, cuspidal, automorphic representation of $G L_{n} (A)$ . The poles are simple and occur at $s = \frac{m _{1} + m _{2}}{4} - \frac{i}{2}$ , $0 \leq i \leq min (m_{1}, m_{2}) - 1$ . Our methods also show that when $m_{1} = m_{2}$ , the above Eisenstein series vanish at s=0.