Supercuspidal Character Survey Heading

Definitions and mathematical statement

The supplied extract concerns complex representations in the character theory of $p$-adic groups and is attributed to Loren Spice under the title Supercuspidal Characters: What and How(e). Its organizing question is: "What do characters look like?" The coefficient field is thus ${\mathbb{C}}$, but the extract specifies no particular group, base field, class of representations, notion of character, evaluation domain, or desired character formula.

Unrecoverable mathematical statement. No precise mathematical proposition or quantified open problem can be recovered from this extract. The supplied statement-verification note identifies the question as an expository section heading and historical organizing question. Accordingly, this draft retains the question as exposition; it does not supply missing hypotheses or assert a conjecture about supercuspidal characters. In particular, the title alone does not determine the intended mathematical target.

The abstract explicitly presents only one strand in the development of modern character theory of $p$-adic groups, rather than a comprehensive history. It says that important work was omitted from the talk and that space limitations required further omissions from the abstract, without intending any slight to omitted contributors. All historical claims are qualified by "to the best of my knowledge," including where that qualification is implicit.

Short English statement

The supplied passage asks broadly what characters of complex representations of $p$-adic groups look like. It is an expository survey heading, according to the supplied verification note, and provides insufficient information for a precise conjecture. The registry's open label is supplied curation, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006121$, title Supercuspidal Character Survey Heading, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies status open, statement status unrecoverable, and a verification note describing the extracted question as an expository heading rather than a precise open problem.
• Loren Spice, Supercuspidal Characters: What and How(e), as attributed in the supplied extract. The registry gives no separate problem proposer; this attribution identifies the abstract's author and does not establish authorship of a conjecture.
• Supplied report citation: Representations of $p$-adic Groups (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/54, pp. 3103--3178. https://doi.org/10.4171/owr/2024/54. The record separately gives proposed year 2024 and the tag owr-year:2024; these supplied year designations have not been reconciled.
• Supplied literature source, labelled "OWR report": https://publications.mfo.de/bitstream/handle/mfo/4270/OWR_2024_54.pdf?sequence=1. The literature assessment dated 2026-08-22 is supplied dated curation, not independently verified here. It labels the item open, characterizes the extracted question as broad rather than a specific theorem, and says that the cited supercuspidal-character literature does not establish its resolution. The supplied background requests independent verification of time-sensitive status and cited claims.