Dual enhanced Shahidi conjecture for unramified Arthur-type representations

Let G\mathrm{G} be a connected reductive group over a non-Archimedean local field kvk_v, and let G=G(kv)G=\mathrm{G}(k_v). Assume that a theory of local Arthur packets for GG exists. For a local Arthur parameter ψΨ+(G)\psi\in\Psi^+(G), let ϕψ\phi_{\psi} be its associated local LL-parameter; an Arthur-type representation is one lying in some local Arthur packet, and an unramified representation is one with a nonzero hyperspecial-fixed vector. Dual enhanced Shahidi conjecture. Every unramified Arthur-type representation of GG lies in exactly one local Arthur packet, and that packet is associated with an anti-generic local Arthur packet. Moreover, if ϕψ\phi_{\psi} is unramified, then Πψ\Pi_{\psi} contains a unique unramified representation, namely the representation associated with ϕψ\phi_{\psi} through the Satake isomorphism. The source says this is proved for quasi-split classical groups and for GnG_n in the paper, while the general connected reductive case remains open.

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Primary source

Alexander Hazeltine, Baiying Liu and Chi-Heng Lo, “On the enhanced Shahidi conjecture and global applications”, arXiv:2404.05773 (2024).

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