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

David Ginzburg, David Soudry

Abstract

We determine the poles of the Eisenstein series on a general linear group, induced from two Speh representations, $Δ(τ,m_1)|\cdot|^s\timesΔ(τ,m_2)|\cdot|^{-s}$, $Re(s)\geq 0$, where $τ$ 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.

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

Abstract

We determine the poles of the Eisenstein series on a general linear group, induced from two Speh representations, , , where is an irreducible, unitary, cuspidal, automorphic representation of . The poles are simple and occur at , . Our methods also show that when , the above Eisenstein series vanish at s=0.

Paper Structure

This paper contains 21 sections, 23 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Nilpotent orbits and Fourier coefficients
  3. Unipotent groups and characters attached to partitions
  4. Fourier coefficients
  5. Degenerate Whittaker coefficients and generalized Whittaker coefficients
  6. Fourier coefficients with respect to $\underline{p}=(k,1^{\ell-k})$
  7. Fourier coefficients with respect to $(k^r)$
  8. A descent operation on Eisenstein series induced from Speh representations
  9. Relation of the $\psi_{(kn,1^{(m-k)n})}$-coefficients on $\rho_{\Delta(\tau,\underline{m}),\underline{s}}$ and derivatives
  10. The descent to ${\mathrm{GL}}_{(m-k)n}({\mathbb {A}})$ of the Eisenstein series $E(f_{\Delta(\tau,\underline{m}),\underline{s}})$
  11. The section $\Lambda(f_{\Delta(\tau,\underline{m}),\underline{s}})$
  12. The case $k=2$
  13. The pole at $s=\frac{m_1+m_2}{4}$
  14. The constant term of $E(f_{\Delta(\tau,\underline{m}),s})$ along $V_{n^m}$
  15. Analytic properties of ${\bf Res_{\underline{s}=\underline{s}^0,\underline{\zeta}=\underline{\zeta}^0}M_{w^0w_{n\cdot(\underline{a},\underline{b})},\underline{s},\underline{\zeta},s}(f_{\underline{s}+s\cdot\underline{1},\underline{\zeta}-s\cdot\underline{1}})}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

The list of poles of the Eisenstein series attached to $\rho_{\Delta(\tau,(m_1,m_2)),s}$, for $Re(s)\geq 0$, is All these poles are simple.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 31 more