Restriction Spectra of $\operatorname{GL}_3$ Maass Forms

Definitions and mathematical statement

A Maass form is understood here as a smooth automorphic eigenfunction of the relevant invariant differential operators, with the automorphy, growth, and normalization conditions still to be specified. Let $F$ be the $\operatorname{GL}_3$ Maass form in the source question, and write $M_3$ for its unspecified automorphic quotient. Let \[ \iota:N\longrightarrow M_3 \] denote the codimension-$2$ submanifold inclusion described in the source as arising from the "upper-triangular embedding" of $\operatorname{GL}_2$. Its restriction is the function \[ F_N(y)=F(\iota(y)),\qquad y\in N. \] The source does not specify the embedding, the quotients, or how central and determinant variables are treated; these symbols therefore record the requested geometry without fixing it.

Let $u$ be a given $\operatorname{SL}_2({\mathbb{Z}})$ Maass form. In the usual scalar-valued modular-surface interpretation, its domain is \[ X=\operatorname{SL}_2({\mathbb{Z}})\backslash\mathbb H, \qquad \mathbb H=\{x+iy:x\in{\mathbb{R}},\ y>0\}, \] where $\operatorname{SL}_2({\mathbb{Z}})$ acts by fractional linear transformations. The source does not specify an identification of $N$ with $X$, or a procedure for obtaining a comparison function $u_N$ on $N$ from $u$.

Source question, with unresolved qualifications explicit. For a $\operatorname{GL}_3$ Maass form $F$ and a given $\operatorname{SL}_2({\mathbb{Z}})$ Maass form $u$, determine their restriction inner product. Once a common domain $N$, a comparison function $u_N$, and a measure $\mu_N$ have been specified, the intended pairing would have the form \[ \langle F_N,u_N\rangle =\int_N F(\iota(y))\,\overline{u_N(y)}\,\mathrm d\mu_N(y), \] provided that this integral exists. More generally, determine the spectral decomposition of the restricted form $F_N$ on the intended restriction space.

The integral is a conditional formulation of the requested quantity, not a supplied evaluation or convergence assertion. A precise spectral-decomposition problem requires the restriction space, measure, spectral operators, and function space to be fixed, together with any necessary treatment of non-square-integrable terms. The source does not specify whether an explicit coefficient formula, an identity involving other invariants, or another characterization is sought. It supplies no theorem-level assertion resolving these choices.

Short English statement

The question asks how a $\operatorname{GL}_3$ Maass form decomposes after restriction to a codimension-$2$ submanifold associated with $\operatorname{GL}_2$, including its pairing with a given $\operatorname{SL}_2({\mathbb{Z}})$ Maass form. The embedding, common domain, measure, and spectral setting require clarification before this becomes a fully specified mathematical problem. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied permanent ID $30001876$, source problem code catalogue entry, title "Restriction Spectra of $\operatorname{GL}_3$ Maass Forms." The supplied record belongs to the Oberwolfach Reports Open Problems collection.
• Supplied report citation: The Analytic Theory of Automorphic Forms (2012), Oberwolfach Reports, pp. 2397--2458, DOI 10.4171/owr/2011/43. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2011/43.
• Attribution and chronology: the supplied proposed year is 2011; no individual proposer is identified. The report citation gives 2012, while the DOI and supplied workshop-year tag refer to 2011. These supplied dates are retained without resolving their bibliographic relationship.
• Supplied verification metadata labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied provenance, not independent verification by this draft.
• Supplied literature assessment, checked on 2026-08-21: status "open"; the extracted sentence requests a spectral decomposition without specifying a theorem-level claim, and no general resolution is identified. The listed checked source is the original OWR report linked above. This assessment is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent review.