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

TL;DR

This paper analyzes the poles of Eisenstein series on general linear groups induced from two Speh representations, identifying their locations and properties, and showing vanishing at zero in certain cases.

Contribution

It precisely determines the poles of Eisenstein series induced from two Speh representations and reveals their behavior at specific points, including vanishing when parameters are equal.

Findings

Poles are simple and located at s=(m_1+m_2)/4 - i/2

Poles occur for 0 ≤ i ≤ min(m_1,m_2)-1

Eisenstein series vanish at s=0 when m_1=m_2

Abstract

We determine the poles of the Eisenstein series on a general linear group, induced from two Speh representations, $\Delta(\tau,m_1)|\cdot|^s\times\Delta(\tau,m_2)|\cdot|^{-s}$, $Re(s)\geq 0$, where $\tau$ is an irreducible, unitary, cuspidal, automorphic representation of $GL_n({\bf A})$. 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.