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Theta correspondence and Arthur packets: on the Adams conjecture

Petar Bakic, Marcela Hanzer

Abstract

The Adams conjecture predicts that the local theta correspondence should respect Arthur packets. In this paper, we revisit the Adams conjecture for the symplectic--even orthogonal dual pair. Our results provide a precise description of all situations in which the conjecture holds.

Theta correspondence and Arthur packets: on the Adams conjecture

Abstract

The Adams conjecture predicts that the local theta correspondence should respect Arthur packets. In this paper, we revisit the Adams conjecture for the symplectic--even orthogonal dual pair. Our results provide a precise description of all situations in which the conjecture holds.
Paper Structure (27 sections, 49 theorems, 201 equations)

This paper contains 27 sections, 49 theorems, 201 equations.

Table of Contents

  1. Preliminaries and Notation
  2. Groups
  3. Witt towers
  4. Parabolic subgroups
  5. Representations
  6. Notation for induction
  7. Segments
  8. Notation for Jacquet functors
  9. The Langlands classification
  10. The theta correspondence
  11. Arthur packets
  12. The Adams conjecture
  13. Examples
  14. Technical results
  15. Mœ glin's construction
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\alpha > 1$ be odd. Assume $\pi_\alpha = \theta_{-\alpha}(\pi)$ and $\pi_{\alpha-2} \neq 0$. Then $\theta_{-(\alpha-2)}(\pi)=\pi_{\alpha-2}$; in particular, $\theta_{-(\alpha-2)}(\pi)\neq 0$ is in $\Pi_{\psi_{\alpha-2}}$.

Theorems & Definitions (108)

  • Conjecture : Adams Adams
  • Theorem 1: Theorem \ref{['theoremA']}
  • Theorem 2: Theorem \ref{['theoremB']}
  • Lemma 1.1
  • proof
  • Theorem 1.2: Howe duality
  • Proposition 1.3: Tower property
  • Remark 1.4
  • Remark 1.6
  • Remark 1.7
  • ...and 98 more