Cuspidal Spectral-Multiplier Theorem

Definitions and mathematical statement

The supplied record concerns a reductive group $G$ over ${\mathbb{Q}}$, a finite set $S$ of primes, and an irreducible cuspidal automorphic representation $\pi$ having nonzero $K$-invariants, written $\pi^K\ne\{0\}$, and satisfying the supplied condition that $\pi$ is not $S$-CAP. The subgroup $K$, the definition of $S$-CAP, the meaning of an $S$-multiplier, and the admissible test functions are not specified in the supplied record.

The asserted spectral-isolation property can therefore be transcribed only schematically. Write $\mathcal H$ for the unspecified ambient automorphic Hilbert space, $\mathcal H_{\mathrm{cusp}}$ for its cuspidal subspace, and $V_\pi\subseteq\mathcal H_{\mathrm{cusp}}$ for a realization of $\pi$. Here $R(f)$ denotes the convolution operator associated with a test function $f$, and $\mu\star f$ denotes the multiplier action written as convolution in the source. These symbols describe the roles required by the supplied claim; they do not resolve the missing analytic definitions.

With those qualifications, the supplied assertion is that for every such $G,S,\pi$, there exists an $S$-multiplier $\mu_\pi$ such that, for every admissible test function $f$, \[ \left.R(\mu_\pi\star f)\right|_{\mathcal H_{\mathrm{cusp}}^{\perp}}=0, \qquad \left.R(\mu_\pi\star f)\right|_{V_\pi} = \left.R(f)\right|_{V_\pi}. \] The orthogonal complement is taken inside $\mathcal H$. The existential quantifier for $\mu_\pi$ precedes the universal quantifier for $f$, so the multiplier is required to work simultaneously for all admissible test functions. The record does not determine whether the second identity is intended on a chosen copy of $\pi$ or its entire isotypic component.

This is a qualified transcription, not a complete theorem statement. In particular, the ambient automorphic quotient, central-character or central-quotient conventions, operator domains, and test-function and multiplier spaces require specification. The supplied registry labels the item open and its statement status unrecoverable. Its separate statement-verification note instead identifies the extraction as Theorem 1 of the cited report, a proved spectral-isolation result rather than an open problem. The earlier supplied literature assessment describes a truncated fragment beginning with "Conjecture", which does not match the displayed assertion. These conflicting descriptions remain unresolved here.

Short English statement

The supplied assertion asks for a multiplier that preserves the action of every admissible test function on a specified non-$S$-CAP cuspidal representation while making that action vanish on the orthogonal complement of the cuspidal spectrum. Essential definitions and analytic conventions are missing. The supplied open label conflicts with a verification note identifying the result as a proved theorem; neither description is independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004350$, title "Cuspidal Spectral-Multiplier Theorem", exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Modular Forms (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/57, pp. 3529--3606. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2019/57. The report title year and DOI year are retained as supplied.
• The literature assessment was checked on 2026-08-22 according to the supplied record. This is supplied dated curation, not an independently verified literature assessment. It classifies the item as open under a rule for unclear records because it describes the extraction as a truncated fragment beginning with "Conjecture".
• The supplied statement-verification note identifies the extraction as Theorem 1 in the official report and describes it as a proved spectral-isolation result for non-$S$-CAP cuspidal representations. It says that no separate clean statement was promoted and that the original record was retained for provenance. This source-identification claim has not been independently verified here.