The Refined Formal-Degree Conjecture

Definitions and mathematical statement

The supplied record calls the following assertion the refined formal-degree conjecture. Write $A_G$ for the split center of the group $G$, and $\deg(\pi,\mu_G)$ for the formal degree of the representation $\pi$ with respect to the measure $\mu_G$. The source also uses a parameter $\phi$, a representation $\rho$, a group $S_\phi$, an invariant $r(G)$, an adjoint representation $\operatorname{Ad}_G$, and a principal parameter $\phi_{\operatorname{prin}}$. Their precise definitions and admissibility conditions are not supplied; these symbols retain the intended source meanings without imposing additional hypotheses.

For the intended data, the proposed identity under the hypothesis $A_G=\{1\}$ is \[ (-1)^{r(G)}\deg(\pi,\mu_G) = \sigma\,\frac{\dim\rho}{|S_\phi|} \frac{\gamma\!\left(0,\operatorname{Ad}_G\circ\phi\right)} {\gamma\!\left(0,\operatorname{Ad}_G\circ\phi_{\operatorname{prin}}\right)}, \qquad \sigma\in\{+1,-1\}. \] Here $\gamma$ denotes the gamma factor appearing in the supplied formula. The source says that $\sigma$ is given by the quotient of the corresponding root numbers. It does not explicitly define that quotient or the normalizations of the root numbers and gamma factors.

This is a conditional transcription requiring further specification, rather than a fully quantified conjecture. In particular, the source does not specify the base field, the class of groups or representations, the relation between $\pi$ and $(\phi,\rho)$, the meaning of $S_\phi$ and $r(G)$, the normalization of $\mu_G$, or the interpretation of the displayed quotient at $s=0$ if its factors are singular or vanish. The intended range of quantification and these conventions must be recovered before a precise mathematical formulation can be accepted.

Short English statement

When the split center is trivial, the conjecture relates a signed formal degree to a representation-dimension factor and a ratio of adjoint gamma factors, with a sign prescribed by root numbers. Essential hypotheses and normalizations are missing from the supplied excerpt. The supplied dated curation labels the problem partially solved and distinguishes a claimed theorem for tempered orthogonal root numbers from the partly resolved refined formal-degree statement; these assessments are not independently verified here.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004796$, source problem code catalogue entry, title The Refined Formal-Degree Conjecture, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2021.
• Supplied source citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, pp. 2089--2156, DOI 10.4171/owr/2021/39. Original Oberwolfach report: https://doi.org/10.4171/owr/2021/39. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Schwein, orthogonal root numbers, https://arxiv.org/abs/2107.02360.
• Supplied literature reference, labeled "published orthogonal-root-number theorem": https://doi.org/10.1007/s00208-022-02416-6.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that the refined central-character identity for orthogonal root numbers is proved for tempered parameters, while the accompanying refined formal-degree statement and its full generality remain only partly resolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, with the explanation that explicit formulas were dollar-delimited and the supplied mathematical content was preserved. This records an editorial check and does not establish mathematical correctness or completeness.