The Arthur-closure conjecture for the unitary dual

Let FF) be a non-Archimedean local field of characteristic zero, let G\mathrm{G} be a connected reductive group over FF, and set G=G(F)G=\mathrm{G}(F). Let Πu(G)\Pi_u(G) denote the unitary dual, let ΠA(G)\Pi_A(G) denote the Arthur representations, and let ΠA(G)\Pi_{\overline{A}}(G) be the closure of ΠA(G)\Pi_A(G) under complementary series and unitary parabolic induction. Arthur-closure conjecture. Assuming the theory of local Arthur packets, the unitary dual is generated by the Arthur representations:

ΠA(G)=Πu(G).\Pi_{\overline{A}}(G)=\Pi_u(G).

This conjecture proposes a construction of the whole unitary dual from Arthur representations and is stated for connected reductive groups over non-Archimedean local fields of characteristic zero. Its status is not resolved in the supplied text.

Sources & referencesView supporting material

Primary source

Alexander Hazeltine, Dihua Jiang, Baiying Liu, Chi-Heng Lo and Qing Zhang, “On the complementary Arthur representations and unitary dual for p-adic classical groups”, arXiv:2505.11381 (2026).

Additional references

3 papers in this index state this conjecture (2011–2025). The statement above is taken from the most recent of them; the others are arXiv:2410.11806, arXiv:1103.0043.

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