Some unipotent Arthur packets for p-adic split F4

TL;DR

This paper investigates unipotent representations of the split p-adic group F4, computes characteristic cycles of associated perverse sheaves, and confirms that weak Arthur packets are unions of micro-packets, advancing understanding of p-adic representation theory.

Contribution

It introduces a detailed analysis of unipotent Arthur packets for split p-adic F4, including computation of characteristic cycles and verification of the union property of weak Arthur packets.

Findings

Computed characteristic cycles of perverse sheaves for unipotent representations.

Verified that weak Arthur packets are unions of micro-packets.

Provided new insights into the structure of Arthur packets for p-adic F4.

Abstract

Let $G(k)$ be the split form of the simple exceptional p-adic group of type $F_4$, and let $\mathcal O = F_4(a_3)$ be the minimal distinguished nilpotent orbit. Our main result concerns the class of unipotent representations with cuspidal support at infinitesimal character $\Lambda$ determined by $\mathcal O$. These representations are parameterized by local systems, $\{(S, \mathcal L)\}$. We compute the characteristic cycles of the perverse sheaves $\text{IC}(S, \mathcal L)$ and determine all micro-packets in the sense of [Vo93]. In [CMBO24], the authors introduced a notion of weak Arthur packets in the p-adic setting. They conjectured that weak Arthur packets are unions of Arthur packets, in an appropriate sense. We verify that weak Arthur packets are unions of micro-packets.