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Local Arthur-packet conjectures for even orthogonal groups

Let GnG_n be an even orthogonal group over a non-Archimedean local field and let ψ\psi be a local Arthur parameter for GnG_n. Determine the local Arthur packet Πψ(Gn)\Pi_\psi(G_n) and, for every irreducible admissible representation π\pi of GnG_n, give a precise criterion for whether π∈Πψ(Gn)\pi\in\Pi_\psi(G_n), in a manner satisfying the conjectural local Arthur-packet description.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Theta-packet formulation in the quasi-split case

    For a quasi-split even orthogonal group GG, identify the theta packet and the conjectural Arthur packet by proving Πψθ(G)=ΠψA(G)\Pi_{\psi}^{\theta}(G)=\Pi_{\psi}^{A}(G) for every relevant local Arthur parameter ψ\psi.

References

Primary source

arXiv

Additional references

automorphic-formsnumber-theoryrepresentation-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

An unrefereed September 2026 paper claims to prove several classification conjectures for even orthogonal groups, but the broader problem is not settled.

The problem concerns conjectural descriptions of local Arthur packets for even orthogonal groups over non-Archimedean fields. No original proposer or date is identified in the supplied sources.

Known results

  • A 2020 preprint constructed a local Langlands correspondence for arbitrary even orthogonal groups via theta lifts, but explicitly left twisted endoscopic character relations unproved.
  • A 2021 preprint claimed independence of theta packets from the dual pair and, in the quasi-split case, the identification Πψθ(G)=ΠψA(G)\Pi_{\psi}^{\theta}(G)=\Pi_{\psi}^{A}(G).
  • A 2026 preprint studied packet multiplicities at first occurrence and reported values exceeding those predicted by the Adams conjecture.

September 23, 2026 claimed progress

Alexander Hazeltine, Baiying Liu, and Chi-Heng Lo report explicit packet and membership algorithms proving several named conjectures in the even orthogonal setting. The result is restricted to that setting and is not yet refereed; it therefore establishes claimed progress, not a verified resolution of the whole entry.

Current status (as of September 2026): Several named conjectures are claimed proved for even orthogonal groups, but the claims are unverified and the broader collection of local Arthur-packet conjectures remains open.

Sources

Solutions 0

No solutions have been posted yet.