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On residual automorphic representations and period integrals for symplectic groups

Solomon Friedberg, David Ginzburg, Omer Offen

Abstract

We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of $\mathrm{GL}_{2n}$ with a linear period to an irreducible component of the residual spectrum of the rank $k$ symplectic group $\mathrm{Sp}_k$ for any $k\ge 2n$. We show that this residual representation admits a non-zero $\mathrm{Sp}_n\times \mathrm{Sp}_{k-n}$-invariant linear form. This generalizes a construction of Ginzburg, Rallis and Soudry, the case $k=2n$, that arises in the descent method.

On residual automorphic representations and period integrals for symplectic groups

Abstract

We construct new irreducible components in the discrete automorphic spectrum of symplectic groups. The construction lifts a cuspidal automorphic representation of with a linear period to an irreducible component of the residual spectrum of the rank symplectic group for any . We show that this residual representation admits a non-zero -invariant linear form. This generalizes a construction of Ginzburg, Rallis and Soudry, the case , that arises in the descent method.

Paper Structure

This paper contains 58 sections, 31 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Parabolic subgroups and Iwasawa decomposition
  4. Roots and Weyl groups
  5. Roots
  6. Weyl groups and Bruhat decomposition
  7. Elementary symmetries
  8. Special notation for general linear groups
  9. Special notation for symplectic groups
  10. The Weyl group
  11. Standard parabolic subgroups of $G$
  12. Reduced Weyl elements
  13. Automorphic forms
  14. Eisenstein series and constant terms
  15. Intertwining operators
  16. ...and 43 more sections

Key Result

Theorem 1.1

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • proof
  • Corollary 4.4
  • proof
  • ...and 50 more