Weak Arthur packets as unions of simplified Arthur packets

From papers

Let Og\mathcal O\subset\mathfrak{g}^{\vee} be a nilpotent orbit, let ψO\psi_{\mathcal O} be the corresponding basic Arthur parameter, and let {e,h,f}\{{\it e},{\it h},{\it f}\} be an sl2\mathfrak{sl}_2-triple attached to O\mathcal O with htr{\it h}\in\mathfrak{t}^{\vee}_r. Let ΠψOweak(G(k))\Pi^{\text{weak}}_{\psi_{\mathcal O}}(\textmd{G}(\textmd{k})) be the associated weak Arthur packet. Let T\mathcal T be a set of simplified Arthur parameters satisfying the source's conditions, and let Πψ~jArt(G(k))\Pi_{\widetilde{\psi}_j}^{\text{Art}}(\textmd{G}(\textmd{k})) denote the corresponding Arthur packets.

Weak Arthur packet decomposition conjecture. There is a set T\mathcal T and an appropriate notion of Arthur packet such that

ΠψOweak(G(k))=ψ~jTΠψ~jArt(G(k)).\Pi^{\text{weak}}_{\psi_{\mathcal O}}(\textmd{G}(\textmd{k}))=\bigcup_{\widetilde{\psi}_j\in\mathcal T}\Pi_{\widetilde{\psi}_j}^{\text{Art}}(\textmd{G}(\textmd{k})).

This is the cited Conjecture 3.1.2 of cmbo, and it gives a precise form of the weak-Arthur-packet union principle. The source supplies no evidence that it has been resolved.

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Sources & referencesView supporting material

Primary source

Leticia Barchini and András C. Lőrincz, “Some unipotent Arthur packets for p-adic split F4”, arXiv:2512.07014 (2025).

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