Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-zero condition on Mœglin-Renard's parametrization for Arthur packets of $\mathrm U(p,q)$

Chang Huang

TL;DR

The paper develops a non-zero criterion for Mœglin-Renard's parametrization of Arthur packets for real unitary groups ${\mathrm U}(p,q)$, presenting a system of linear-type constraints that mirror the $p$-adic criterion of Atobe and Trapa. It introduces a robust framework based on admissible segment arrangements, transition maps between parameter spaces, and a refined Trapa-style tableau analysis to ensure the criterion is both necessary and sufficient. By aligning the real and $p$-adic formalisms, the work suggests a map between Arthur-Vogan packets across the real and $p$-adic settings, potentially enabling a unified perspective on endoscopic classification. The results provide concrete, checkable conditions for non-vanishing of $A_{\mathfrak q}(\lambda)$ and illuminate how cohomological induction interacts with parametrization under various admissible orders. Overall, the approach enhances explicit understanding of multiplicity-one phenomena and deepens connections between real and $p$-adic Arthur packets.

Abstract

Mœglin-Renard parametrized A-packet of unitary group through cohomological induction in good parity case. Each parameter gives rise to an $A_{\mathfrak q}(λ)$ which is either $0$ or irreducible. Trapa proposed an algorithm to determine whether a ``mediocre'' $A_{\mathfrak q}(λ)$ of $\mathrm U(p, q)$ is non-zero. Based on his result, we present a further understanding of the non-zero condition on Mœglin-Renard's parametrization. Our criterion comes out to be a system of linear constraints, and has the same formulation as $p$-adic case. This suggests a map from A-packets of real unitary group to A-packets of $p$-adic symplectic group or special orthogonal group.

Non-zero condition on Mœglin-Renard's parametrization for Arthur packets of $\mathrm U(p,q)$

TL;DR

The paper develops a non-zero criterion for Mœglin-Renard's parametrization of Arthur packets for real unitary groups , presenting a system of linear-type constraints that mirror the -adic criterion of Atobe and Trapa. It introduces a robust framework based on admissible segment arrangements, transition maps between parameter spaces, and a refined Trapa-style tableau analysis to ensure the criterion is both necessary and sufficient. By aligning the real and -adic formalisms, the work suggests a map between Arthur-Vogan packets across the real and -adic settings, potentially enabling a unified perspective on endoscopic classification. The results provide concrete, checkable conditions for non-vanishing of and illuminate how cohomological induction interacts with parametrization under various admissible orders. Overall, the approach enhances explicit understanding of multiplicity-one phenomena and deepens connections between real and -adic Arthur packets.

Abstract

Mœglin-Renard parametrized A-packet of unitary group through cohomological induction in good parity case. Each parameter gives rise to an which is either or irreducible. Trapa proposed an algorithm to determine whether a ``mediocre'' of is non-zero. Based on his result, we present a further understanding of the non-zero condition on Mœglin-Renard's parametrization. Our criterion comes out to be a system of linear constraints, and has the same formulation as -adic case. This suggests a map from A-packets of real unitary group to A-packets of -adic symplectic group or special orthogonal group.
Paper Structure (46 sections, 36 theorems, 184 equations)

This paper contains 46 sections, 36 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Notations and results
  3. A-parameter
  4. Construction and parametrization of A-packet
  5. Admissible representatives of infinitesimal character
  6. Admissible action on the generalized parameter space
  7. Our non-zero criterion
  8. Framework of proof
  9. Comparison with the $p$-adic side
  10. A-parameter
  11. Arthur-Vogan packet
  12. Parameter space
  13. Transition map
  14. Basic necessary condition
  15. Sufficient condition
  16. ...and 31 more sections

Key Result

Theorem 2.1

Denote the A-packet of $G$ attached to $\psi$ by $\Pi_\psi(G)$. There is a bijection: where $G_{\mathrm g} = {\mathrm U}(p- n_{\mathrm b}', q- n_{\mathrm b}')$. In particular, if $\Pi_{\psi_{\mathrm g}}(G_{\mathrm g})$ is multiplicity free, then so is $\Pi_\psi(G)$.

Theorems & Definitions (96)

  • Theorem 2.1: MR
  • Definition 2.2
  • Remark 2.2.1
  • Theorem 2.3: MR
  • Definition 2.4
  • Definition 2.5
  • Remark 2.3.1
  • Definition 2.6
  • Remark 2.3.2
  • Lemma 2.7
  • ...and 86 more