Distinguished Arthur Parameters and Euler Products of Periods

Definitions and mathematical statement

The source poses two related questions about a symmetric variety, but does not supply enough hypotheses to specify a fully quantified conjecture. The following notation makes the intended questions explicit while retaining those omissions.

Let $F$ denote the intended global field, $\mathbb A_F$ its adele ring, and $F_v$ its completion at a place $v$. The source does not specify the type or characteristic of $F$. Write $G$ for the ambient algebraic group, $H$ for the symmetric subgroup, and $X=H\backslash G$, with $X_v=H(F_v)\backslash G(F_v)$. This quotient convention is used to describe the questions; the precise class of groups and the relation of $H$ to the fixed points of an involution are not specified in the record.

For an automorphic representation admitting a restricted tensor product decomposition $\pi\simeq\bigotimes_v'\pi_v$, put \[ \mathcal M_v(\pi_v)=\operatorname{Hom}_{H(F_v)}(\pi_v,\mathbb C). \] Here the intended representation spaces and continuity requirements on the linear functionals must still be specified. Interpreting a global $H$-invariant functional adelically gives the analogous space \[ \mathcal M(\pi)=\operatorname{Hom}_{H(\mathbb A_F)}(\pi,\mathbb C). \] This interpretation is a template: the source does not define the global functional or identify it with a particular convergent or regularized automorphic period.

The first question is whether, under the source's unspecified "suitable multiplicity-one hypotheses," a given nonzero global invariant functional $\ell$ factors through the local invariant functionals selected by the relevant Plancherel formulas. Schematically, the requested identity is \[ \ell\!\left(\bigotimes_v f_v\right) = C\prod_v\ell_v(f_v), \qquad C\in\mathbb C^\times, \quad 0\ne\ell_v\in\mathcal M_v(\pi_v), \] for decomposable vectors in its intended domain. The scalar $C$ records the normalization freedom of the local functionals. The source does not specify the multiplicity-one assumptions, the local Plancherel functionals, their normalizations, the measures, or the meaning and convergence of the product. In particular, uniqueness of local invariant functionals does not by itself specify the asserted Euler product.

For the second question, suppose that the intended measures and unitary action defining $L^2(X_v)$ have been fixed, and write $\mu_{X_v}$ for its Plancherel spectral measure on the unitary dual $\widehat{G(F_v)}$. The requested containment is schematically \[ \operatorname{supp}(\mu_{X_v}) \subseteq \left\{\sigma\in\widehat{G(F_v)}: \sigma\text{ has an }X\text{-distinguished Arthur parameter}\right\}. \] This is asked at the places covered by the intended local theory, which the source does not identify. The source also supplies no definition of an $X$-distinguished Arthur parameter or rule relating parameters to representations or packets. Consequently the right-hand side remains an unresolved source-dependent predicate. Whether "support" means topological support or concentration of the measure up to a null set is likewise unspecified. These formulas are statement schemata requiring those qualifications to be resolved, rather than a complete conjecture with silently chosen hypotheses.

Short English statement

The problem asks whether suitable multiplicity-one assumptions make a global invariant functional factor as an Euler product of the local functionals occurring in Plancherel formulas, and whether the local Plancherel spectrum has only $X$-distinguished Arthur parameters. The supplied record leaves the admissible setting, normalizations, parameter condition, and meaning of spectral support unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30001107$, exact source problem code catalogue entry, title "Distinguished Arthur Parameters and Euler Products of Periods," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The supplied proposed year is $2008$.
• Supplied report citation: Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups (2009), Oberwolfach Reports, pp. 2985--3028, DOI 10.4171/owr/2008/53; https://doi.org/10.4171/owr/2008/53. The background describes this collection as extracted from mathematical workshop reports and open-problem discussions. No precise page, speaker, or verbatim report passage for this problem is supplied. The citation's $2009$ label and its $2008$ DOI component are retained as supplied.
• Supplied literature reference, labeled "Sakellaridis--Venkatesh": https://arxiv.org/abs/1203.0039.
• Supplied literature reference, labeled "explicit symmetric-space case": https://arxiv.org/abs/1910.09026.
• Supplied literature reference, labeled "Hermitian-matrix case": https://doi.org/10.1016/j.jnt.2021.06.001.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and describes the general support conjecture as a Sakellaridis--Venkatesh conjecture, with Plancherel formulas and predicted parameter support established in substantial families, including several symmetric and low-rank spaces. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and describes its processing as meaning-preserving semantic, typographic, and LaTeX cleanup without unsupported source-dependent additions. These are supplied registry qualifications, not mathematical verification of this draft. No linked source has been independently consulted here.