Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups

Ron Erez

Abstract

This work is largely inspired by the 2003 Ph.D. thesis \cite{snitz} of Kobi Snitz. In his thesis, Snitz constructed two irreducible, automorphic, cuspidal representations $ π$ and $ π' $ of the metaplectic group $ G\left ( \mathbb A \right ) = \widetilde{ SL }_{ 2 } \left ( \mathbb A \right ) $ where each representation is obtained from a different global theta lifts of certain non-trivial automorphic characters $ ξ$ and $ ξ' $ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V' \right ) \left ( \mathbb A \right ) $, respectively, where $ \mathbb A = \mathbb A_{ \mathbb F } $ is the adele ring of a number field $ \mathbb F $. Snitz shows that for certain matching data of quadratic spaces and automorphic quadratic characters, that these two representations of $ G \left ( \mathbb A \right ) $ are isomorphic, i.e. $ π\congπ'$. The goal of this work is to reformulate and generalize Snitz's work to higher rank groups. Namely we wish to determine for which admissible data $\left ( \left ( q, V \right ) ,ξ, \left ( q', V' \right ),ξ'\right )$ satisfying certain local necessary conditions could an isomorphism possibly exist between two global theta lifts $ π$ and $ π'$ with respect to two reductive dual pairs $ H_{ \mathbb A } \times G_{ \mathbb A } $ and $ H'_{ \mathbb A } \times G_{ \mathbb A } $ and two non-trivial automorphic quadratic characters $ ξ$ and $ ξ'$ of the orthogonal groups $ H_{ \mathbb A } = O \left ( q, V \right ) \left ( \mathbb A \right ) $ and $ H_{ \mathbb A } '= O \left ( q', V'\right ) \left ( \mathbb A \right ) $, respectively and the group $ G $ which is the symplectic or the metaplectic group.

Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups

Abstract

This work is largely inspired by the 2003 Ph.D. thesis \cite{snitz} of Kobi Snitz. In his thesis, Snitz constructed two irreducible, automorphic, cuspidal representations and of the metaplectic group where each representation is obtained from a different global theta lifts of certain non-trivial automorphic characters and of the orthogonal groups and , respectively, where is the adele ring of a number field . Snitz shows that for certain matching data of quadratic spaces and automorphic quadratic characters, that these two representations of are isomorphic, i.e. . The goal of this work is to reformulate and generalize Snitz's work to higher rank groups. Namely we wish to determine for which admissible data satisfying certain local necessary conditions could an isomorphism possibly exist between two global theta lifts and with respect to two reductive dual pairs and and two non-trivial automorphic quadratic characters and of the orthogonal groups and , respectively and the group which is the symplectic or the metaplectic group.
Paper Structure (91 sections, 49 theorems, 292 equations)

This paper contains 91 sections, 49 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The global theta correspondence in a nutshell
  4. An example of lifting a trivial automorphic character
  5. An example of lifting a non-trivial automorphic quadratic character
  6. Goal of this work
  7. Simplifying Assumptions
  8. Main Statement
  9. Main Statement in terms of pairings
  10. Overview of proof
  11. Local necessary conditions and admissible quadruples
  12. Local necessary conditions
  13. An example of a locally admissible quadruple in the low rank setting
  14. Determining isomorphisms via global and local pairings
  15. The Global Pairings
  16. ...and 76 more sections

Key Result

Theorem 1.1

Given a globally admissible quadruple $\alpha = \left ( \left ( q, V \right ) ,\xi , \left ( q ^{\prime} , V ^{\prime} \right ),\xi ^{\prime} \right )$. If a certain global partial $L$-function $\Delta_{ S }$ has a pole at $s = \rho_{ n } := \frac{ n+1 }{ 2 }$ then the global theta lifts are isomorphic.

Theorems & Definitions (134)

  • Theorem 1.1: Main statement
  • Theorem 1.2: Alternative main statement in terms of pairings
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.2: Example of a locally admissible quadruple
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7: The local pairing
  • ...and 124 more