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Vogan's Conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_n$ and a combinatorial Lemma

Chi-Heng Lo

TL;DR

The work delivers a new, elementary proof of the generalized combinatorial lemma underpinning Vogan's conjecture for $GL_n(F)$, using only the Mœglin–Waldspurger algorithm. By reframing Arthur-type multi-segments and exploiting a structured reduction (Proposition main) that isolates a distinguished $oldsymbol{ au}_{d,a}$ block, the authors execute an induction on the number of constituents to show that dominance and tilde-dominance force equality. This yields an alternative route to establishing the Vogan correspondence between local Arthur packets and ABV-packets for $GL_n(F)$ and strengthens the link between MW algorithm, Zelevinsky/Pyasetskii involutions, and combinatorial multi-segment analysis. The results enrich the combinatorial toolkit for understanding local packets in the $p$-adic setting and complement endoscopic approaches via a self-contained, algorithmic perspective.

Abstract

For $\mathrm{GL}_n$ over a $p$-adic field, Cunningham and Ray proved Vogan's conjecture, that is, local Arthur packets are the same as ABV packets. They used the endoscopic theory to reduce the general case to a combinatorial lemma for irreducible local Arthur parameters, and their proof implies that one can also prove Vogan's conjecture for $p$-adic $\mathrm{GL}_n$ by proving a generalized version of this combinatorial lemma. Riddlesden recently proved this generalized lemma. In this paper, we give a new proof of it, which has its own interest.

Vogan's Conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_n$ and a combinatorial Lemma

TL;DR

The work delivers a new, elementary proof of the generalized combinatorial lemma underpinning Vogan's conjecture for , using only the Mœglin–Waldspurger algorithm. By reframing Arthur-type multi-segments and exploiting a structured reduction (Proposition main) that isolates a distinguished block, the authors execute an induction on the number of constituents to show that dominance and tilde-dominance force equality. This yields an alternative route to establishing the Vogan correspondence between local Arthur packets and ABV-packets for and strengthens the link between MW algorithm, Zelevinsky/Pyasetskii involutions, and combinatorial multi-segment analysis. The results enrich the combinatorial toolkit for understanding local packets in the -adic setting and complement endoscopic approaches via a self-contained, algorithmic perspective.

Abstract

For over a -adic field, Cunningham and Ray proved Vogan's conjecture, that is, local Arthur packets are the same as ABV packets. They used the endoscopic theory to reduce the general case to a combinatorial lemma for irreducible local Arthur parameters, and their proof implies that one can also prove Vogan's conjecture for -adic by proving a generalized version of this combinatorial lemma. Riddlesden recently proved this generalized lemma. In this paper, we give a new proof of it, which has its own interest.
Paper Structure (8 sections, 12 theorems, 98 equations)

This paper contains 8 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A partial ordering on multi-segments
  4. Mœglin-Waldspurger algorithm
  5. Rephrasing Mœglin-Waldspurger algorithm
  6. Rephrasing Lemma \ref{['lem main intro']} by multi-segments
  7. Preparation for reduction
  8. Proof of Lemma \ref{['lem main intro']}

Key Result

Lemma 1.2

Let $\psi$ be an irreducible local Arthur parameter of $\mathrm{GL}_n(F)$ and denote $\phi_{\psi}$ the associated $L$-parameter. If $\phi$ is an $L$-parameter of $\mathrm{GL}_n(F)$ satisfying that $\phi \geq \phi_{\psi}$ and $\widehat{\phi} \geq \widehat{\phi_{\psi}}$, then $\phi=\phi_{\psi}$.

Theorems & Definitions (23)

  • Conjecture 1.1: CFMMX22
  • Lemma 1.2: CR22
  • Lemma 1.3
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.4
  • Lemma 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • ...and 13 more