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On the Muić conjecture--the irreducibility of the big theta lift

Marcela Hanzer

TL;DR

The paper resolves Muić’s conjecture by proving that, for irreducible discrete series $\pi$ over a non-archimedean field, the big theta lift $\Theta(\pi)$ is irreducible whenever non-zero, in the setting of symplectic–even orthogonal dual pairs. The authors develop an inductive approach based on derivatives of $\pi$ and its theta lift, organizing the analysis via Jacquet modules and Arthur–Adams parameterizations, and carefully exclude all potential non-tempered subquotients. They also describe precisely when theta lifts of tempered representations are irreducible, providing a complete picture across going-down and going-up towers. This advances the understanding of local theta correspondence, connects with the Arthur framework, and has implications for global and relative Langlands program contexts.

Abstract

Let $F$ be a non-archimedean local field of characteristic zero. We study theta correspondence for (complex) representations of symplectic--even orthogonal dual reductive pairs over $F;$ more specifically, the big theta lifts. We prove that, starting from a discrete series representation $π$ of a symplectic (even orthogonal group) over $F,$ its big theta lift $Θ(π)$ (as a representation of an even orthogonal (symplectic) group) if non-zero, is an irreducible representation, thus proving a conjecture of Muić. Building upon this result, we completely describe the situations in which the theta lifts of tempered representations are irreducible and when they are not.

On the Muić conjecture--the irreducibility of the big theta lift

TL;DR

The paper resolves Muić’s conjecture by proving that, for irreducible discrete series over a non-archimedean field, the big theta lift is irreducible whenever non-zero, in the setting of symplectic–even orthogonal dual pairs. The authors develop an inductive approach based on derivatives of and its theta lift, organizing the analysis via Jacquet modules and Arthur–Adams parameterizations, and carefully exclude all potential non-tempered subquotients. They also describe precisely when theta lifts of tempered representations are irreducible, providing a complete picture across going-down and going-up towers. This advances the understanding of local theta correspondence, connects with the Arthur framework, and has implications for global and relative Langlands program contexts.

Abstract

Let be a non-archimedean local field of characteristic zero. We study theta correspondence for (complex) representations of symplectic--even orthogonal dual reductive pairs over more specifically, the big theta lifts. We prove that, starting from a discrete series representation of a symplectic (even orthogonal group) over its big theta lift (as a representation of an even orthogonal (symplectic) group) if non-zero, is an irreducible representation, thus proving a conjecture of Muić. Building upon this result, we completely describe the situations in which the theta lifts of tempered representations are irreducible and when they are not.

Paper Structure

This paper contains 3 sections, 15 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main result

Key Result

Proposition 2.1

For each $l\ge 1,$$\theta_{-l}^{down}(\pi)$ is an unitarizable representation of Arthur type. For each $l\ge l(\pi)+2,$$\theta_{-l}^{up}(\pi)$ is an unitarizable representation of Arthur type. Moreover, if $\phi_{\pi}$ is the Langlands parameter of $\pi,$ then $\theta_{-l}(\pi)$ (on the going-down o Here, we understand each summand $\rho\otimes S_a$ in $\phi_{\pi}$ as of form $\rho\otimes S_a\otim

Theorems & Definitions (30)

  • Conjecture 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 20 more