Some unipotent Arthur packets for p-adic split 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 $\hat{I}›$ 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.