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Functoriality and the theta correspondence

Alexander Hazeltine

Abstract

We study the functoriality of the local theta correspondence for classical $p$-adic groups. This is realized via the adaptation of the Adams conjecture to ABV-packets. We provide evidence for the conjecture, especially in the case of general linear groups.

Functoriality and the theta correspondence

Abstract

We study the functoriality of the local theta correspondence for classical -adic groups. This is realized via the adaptation of the Adams conjecture to ABV-packets. We provide evidence for the conjecture, especially in the case of general linear groups.

Paper Structure

This paper contains 13 sections, 37 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Theta Correspondence
  4. Local Arthur packets
  5. The Adams Conjecture for local Arthur packets
  6. ABV-packets
  7. The Adams conjecture for ABV-packets
  8. The Adams conjecture for general linear groups
  9. Representation theory
  10. Perverse Sheaves
  11. Conormal bundles
  12. The fixed point formula
  13. Proof of the fixed point formula

Key Result

Lemma 1.3

If $H$ is the "going-down" tower (see §sec Theta Correspondence) for $\pi,$ then $\theta(\pi)\in\Pi_{(\phi_\pi)'}^{\mathrm{ABV}}.$$\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (90)

  • Conjecture 1.1: The Adams Conjecture (Ada89)
  • Conjecture 1.2: The Adams Conjecture for ABV-packets
  • Lemma 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 2.1: Howe Duality
  • Theorem 2.2: Conservation relation, SZ15
  • Conjecture 2.3: HLLZ25
  • Conjecture 2.4: The (naive) Adams Conjecture Ada89
  • Theorem 2.5
  • ...and 80 more