Vogan's conjecture on local Arthur packets of $p$-adic $\mathrm{GL}_{n}$ and a combinatorial lemma

Chi-Heng Lo

Vol. 333 (2024), No. 2, 331–356

DOI: 10.2140/pjm.2024.333.331

Abstract

For ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$ 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 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 ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$ by proving a generalized version of this combinatorial lemma. Riddlesden recently proved this generalized lemma. We give a new proof of it, which has its own interest.

Keywords

local Arthur packets, local Arthur parameters, Vogan's conjecture

Mathematical Subject Classification

Primary: 11F70, 22E50

Secondary: 11F85

Milestones

Received: 9 January 2024

Revised: 16 December 2024

Accepted: 22 December 2024

Published: 28 December 2024