Character Bounds beyond Split Levi Subgroups

Definitions and mathematical statement

For an endomorphism $F:G\to G$, write \[ G^F=\{g\in G:F(g)=g\}. \] If $G^F$ is finite, the conventional ordinary-character interpretation of $\operatorname{Irr}(G^F)$ is the set of characters of irreducible finite-dimensional complex representations of $G^F$. In this interpretation, a character bound typically concerns $\lvert\chi(g)\rvert$. These conventions explain the notation; the supplied record does not specify the ambient algebraic group, the endomorphism $F$, the character field, or the precise inequalities intended.

The supplied problem asks the following two questions.

1. Can the character-bound theorem referred to in the source, currently available for split Levi subgroups, be extended to non-split Levi subgroups?
2. When $G$ is simple and is not of type $A$, can useful character bounds be obtained for every pair \[ (g,\chi)\in G^F\times\operatorname{Irr}(G^F)? \]

The first question refers to a theorem whose statement, hypotheses, and bound are absent from the supplied record. The meanings of split and non-split Levi subgroups, and of simplicity for $G$, also require the source context. The second question includes every element and every irreducible character, but does not define what constitutes a useful bound or specify its allowed parameters and dependence on them. Thus the record supplies research targets rather than a fully specified quantitative conjecture; these missing qualifications cannot be recovered from the supplied text alone.

Short English statement

The problem asks whether a character-bound theorem for split Levi subgroups extends to non-split Levi subgroups, and whether useful character bounds hold for every element and every irreducible character when $G$ is simple and not of type $A$. The referenced theorem, ambient hypotheses, and required strength of the bounds are unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004112$, entitled Character Bounds beyond Split Levi Subgroups; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2019; no individual proposer is identified.
• Supplied report citation: Representations of Finite Groups (2020), Oberwolfach Reports, pp. 841--895, DOI 10.4171/owr/2019/14. Supplied source URL and literature reference, labelled "Oberwolfach representation theory report": https://doi.org/10.4171/owr/2019/14. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No specific page, speaker, or theorem number is supplied for these questions.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that no general resolution was found for extensions to non-split Levi subgroups and uniform character bounds beyond established simple type-$A$ and restricted cases. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.