Geometric Description of Pure Local Arthur Packets

Definitions and mathematical statement

The supplied record concerns a group denoted by $G$ over a field denoted by $F$, but does not specify their admissible class. Its notation $\Pi^{\mathrm{pure}}(G/F)$ denotes the collection of representation labels $[\pi,\delta]$ across the pure forms under consideration. The equivalence relation on these labels and the precise meaning of the pure-form datum $\delta$ are not supplied. For an Arthur parameter $\psi$ of $G$, the notation $\Pi^{\mathrm{pure},\psi}(G/F)$ denotes the pure local Arthur packet whose geometric description is sought. The symbols $P(\pi,\delta)$, $\operatorname{Ev}^{0}_{\psi}$, and $A_{\psi}$ denote, respectively, the associated geometric object, the evaluation operation, and the group acting on the resulting object in the intended construction; their definitions and normalization conventions are absent from the supplied record.

For readability, put \[ V_{\psi}(\pi,\delta):=\operatorname{Ev}^{0}_{\psi}\left(P(\pi,\delta)\right). \] The proposed trace identity presupposes that this object carries an $A_{\psi}$-action for which the displayed trace is defined. This is a requirement of the intended formulation, not a construction established by the supplied record.

Proposed statement. For every Arthur parameter $\psi$ of $G$, is it true that \[ \Pi^{\mathrm{pure},\psi}(G/F) = \left\{[\pi,\delta]\in\Pi^{\mathrm{pure}}(G/F): V_{\psi}(\pi,\delta)\neq 0\right\}, \] and that, for every $[\pi,\delta]\in\Pi^{\mathrm{pure},\psi}(G/F)$ and every $a\in A_{\psi}$, \[ \langle a,[\pi,\delta]\rangle_{\psi} = \operatorname{Trace}\!\left(V_{\psi}(\pi,\delta)(a)\right)? \] Here the pairing on the left is the intended Arthur-packet pairing, whose normalization is likewise unspecified. These identities retain the supplied mathematical target, but a standalone precise conjecture requires the missing hypotheses on $F$ and $G$, the parameter conventions, and the definitions and normalizations of the packet, geometric construction, action, and pairing. The supplied literature assessment describes a universal target involving all pure forms and reductive groups; it does not provide the hypotheses needed to formalize that scope.

Short English statement

The proposed identities characterize membership in a pure local Arthur packet by nonvanishing of a geometric evaluation and recover the packet pairing as the trace of the resulting group action. The source leaves the field, group hypotheses, and geometric and normalization conventions unspecified, so this is a qualified statement-only draft requiring mathematical review. The supplied label partially_solved is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied record for permanent ID $30003513$, source problem code catalogue entry, title Geometric Description of Pure Local Arthur Packets; collection: Oberwolfach Reports Open Problems. The project identifies data/unsolvedmath/problems.json as its registry and catalog/problems.json as its supplementary catalog. No individual proposer is identified in the supplied record.
• Supplied report citation: Harmonic Analysis and the Trace Formula (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/25, pp. 1551--1630. https://doi.org/10.4171/owr/2017/25. The record also gives proposed year 2017 and the tag owr-year:2017; the relationship between these dates is not resolved here.
• Supplied literature reference: Cunningham et al., geometric packets. https://arxiv.org/abs/1705.01885.
• Supplied literature reference: Cunningham and Ray, $\mathrm{GL}_n$ case. https://arxiv.org/abs/2302.10300.
• Supplied literature reference: original OWR report. https://doi.org/10.4171/owr/2017/25.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the geometric Arthur-packet identity is proved in important special cases, including $\mathrm{GL}_n$ and split odd special orthogonal groups, and has been computed in examples, while no theorem establishes the universal statement for all pure forms and reductive groups. This is supplied dated curation derived from the OWR collection, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The supplied record labels its statement corrected_verified and reports that the official Oberwolfach report was checked directly and that definitions were restored in the clean statement. These are source-curation claims, not verification performed for this draft; the supplied clean statement does not contain the definitions needed for a standalone formulation.