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Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

Taiwang Deng, Chang Huang, Bin Xu, Qixian Zhao

Abstract

We establish an explicit correspondence of certain Arthur packets between real unitary groups and $p$-adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the $p$-adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the $p$-adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and $p$-adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and $p$-adic sides.

Relating Arthur packets of real unitary groups and $p$-adic symplectic and orthogonal groups

Abstract

We establish an explicit correspondence of certain Arthur packets between real unitary groups and -adic symplectic or orthogonal groups. This allows one to compute Arthur packets of real unitary groups by translating results from the -adic side. A main ingredient in our proof is an explicit relation between Zuckerman's translation functor on the real side and the Jacquet functor on the -adic side. To achieve this, we construct a correspondence of stacks of Langlands parameters with fixed infinitesimal characters between the relevant real and -adic groups. Our approach also allows one to relate the Kazhdan-Lusztig polynomials and the microlocal geometry between real and -adic sides.
Paper Structure (28 sections, 46 theorems, 330 equations)

This paper contains 28 sections, 46 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of proof
  4. Working hypotheses concerning the Local Langlands correspondence
  5. Acknowledgement
  6. Geometry
  7. Langlands correspondence for $U(p,q)$
  8. Langlands correspondence for $p$-adic symplectic/orthogonal groups
  9. The full rank part and the geometric comparison
  10. Explicit correspondence
  11. Proof of Theorem \ref{['thm: comparison of A-packets']} and \ref{['thm: comparison of endoscopic maps']}
  12. Regular case
  13. Singular case
  14. Comparison of translation and derivative
  15. Factorization of translation functors
  16. ...and 13 more sections

Key Result

Theorem 1.5

When $H$ is orthogonal, $\tilde{\iota}$ restricts to a bijection $\Pi_{\psi^{\mathbb{R}}}^{\rm pure}(G) \xrightarrow{\simeq} \Pi_{\psi^{\mathbb{Q}_p}}^{\rm pure}(H)$. When $H$ is symplectic, $\tilde{\iota}$ restricts to a bijection

Theorems & Definitions (81)

  • Theorem 1.5
  • Theorem 1.9
  • Remark 1.10
  • Theorem 1.11
  • Proposition 2.2: ABV:1992, AdC:2009
  • Theorem 2.4: ABV:1992
  • Lemma 2.7
  • proof
  • Lemma 2.10
  • proof
  • ...and 71 more