Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Adams conjecture and intersections of local Arthur packets

Alexander Hazeltine

Abstract

The Adams conjecture states that the local theta correspondence sends a local Arthur packet to another local Arthur packet. Mœglin confirmed the conjecture when lifting to groups of sufficiently high rank and also showed that it fails in low rank. Recently, Bakić and Hanzer described when the Adams conjecture holds in low rank for a representation in a fixed local Arthur packet. However, a representation may lie in many local Arthur packets and each gives a minimal rank for which the Adams conjecture holds. In this paper, we study the interplay of intersections of local Arthur packets with the Adams conjecture.

The Adams conjecture and intersections of local Arthur packets

Abstract

The Adams conjecture states that the local theta correspondence sends a local Arthur packet to another local Arthur packet. Mœglin confirmed the conjecture when lifting to groups of sufficiently high rank and also showed that it fails in low rank. Recently, Bakić and Hanzer described when the Adams conjecture holds in low rank for a representation in a fixed local Arthur packet. However, a representation may lie in many local Arthur packets and each gives a minimal rank for which the Adams conjecture holds. In this paper, we study the interplay of intersections of local Arthur packets with the Adams conjecture.
Paper Structure (15 sections, 36 theorems, 137 equations)

This paper contains 15 sections, 36 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Representations
  4. Theta Correspondence
  5. Local Arthur packets for symplectic groups
  6. Local Arthur packets for orthogonal groups
  7. Raising operators
  8. Mœglin's parameterization
  9. The Adams conjecture
  10. Reduction to good parity
  11. Xu's nonvanishing algorithm
  12. Reduction to chi
  13. Obstructions and proof of Theorem \ref{['thm main thm']}
  14. Obstructions
  15. Proof of Theorem \ref{['thm main thm']}

Key Result

Theorem 1.2

Suppose that $\pi\in\Pi_\psi$ for some $\psi\in\Psi(G).$ Then Conjecture conj Adams intro holds if and only if $\alpha\geq d(\pi,\psi).$

Theorems & Definitions (58)

  • Conjecture 1.1: The Adams conjecture, Ada89
  • Theorem 1.2: BH22
  • Theorem 1.3
  • Theorem 1.4: HLL22
  • Theorem 1.5
  • Lemma 2.1: Frobenius Reciprocity
  • Theorem 2.2: Howe Duality
  • Theorem 2.3: Conservation relation, SZ15
  • Theorem 2.4: Moe06a, Xu17
  • Theorem 2.5: Moe11b
  • ...and 48 more