Tadić's critical-type conjecture for classical groups

Let GG be a classical group over a non-Archimedean local field FF of characteristic zero. Let Πcrit(G)\Pi_{crit}(G) be the representations of critical type, let ΠA,crit(G)=ΠA(G)Πcrit(G)\Pi_{A,\,crit}(G)=\Pi_A(G)\cap\Pi_{crit}(G), let Πu,crit(G)=Πu(G)Πcrit(G)\Pi_{u,\,crit}(G)=\Pi_u(G)\cap\Pi_{crit}(G), and let Πiso(G)\Pi_{iso}(G) be the isolated unitary representations.

Tadić's critical-type conjecture.

ΠA,crit(G)=Πu,crit(G),\Pi_{A,\,crit}(G)=\Pi_{u,\,crit}(G),

and

Πiso(G)Πu,crit(G).\Pi_{iso}(G)\subseteq\Pi_{u,\,crit}(G).

The conjecture seeks to characterize unitary representations of critical type as precisely the Arthur representations of critical type and places all isolated unitary representations in this class. The supplied text gives no resolution of either assertion.

Sources & referencesView supporting material

Primary source

Alexander Hazeltine, Dihua Jiang, Baiying Liu, Chi-Heng Lo and Qing Zhang, “On Arthur representations and the unitary dual”, arXiv:2410.11806 (2026).

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