Unified Constructions of Supercuspidal Representations

Definitions and mathematical statement

For terminology only, let $F$ be a nonarchimedean local field of residue characteristic $p$, and let $G$ be a connected reductive algebraic group over $F$. Under the usual complex-coefficient convention, a representation $(\pi,V)$ of $G(F)$ is smooth if every vector has an open stabilizer. An irreducible smooth representation $\pi$ is supercuspidal if it is not a subquotient of any normalized parabolically induced representation $\operatorname{Ind}_{P(F)}^{G(F)}\sigma$, where $P$ is a proper $F$-parabolic subgroup with Levi factor $M$, and $\sigma$ is an irreducible smooth complex representation of $M(F)$.

Requested construction, with unresolved scope. Does there exist a common construction of supercuspidal representations of reductive $p$-adic groups that simultaneously generalizes the Bushnell--Kutzko and Yu constructions?

The supplied record does not specify the class of fields or groups, the coefficient field, the permitted residue characteristics, any tameness or other hypotheses, the input data of the proposed construction, or the precise sense in which it must recover either existing construction. The conventions above explain standard terminology; they do not supply missing hypotheses for the requested construction. In particular, the record does not determine whether all supercuspidal representations must be obtained or only the representations covered by the two named constructions. Consequently, this is a construction problem with unresolved quantifiers and compatibility requirements, rather than a fully specified mathematical assertion.

Short English statement

Find a common framework that recovers both the Bushnell--Kutzko and Yu constructions of supercuspidal representations. The supplied record leaves the scope and meaning of this generalization unspecified, so a precise existence assertion requires further source clarification.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30004253$, title Unified Constructions of Supercuspidal Representations, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns proposed year $2019$ and names no individual proposer.
• Supplied report citation: New Developments in Representation Theory of p-adic Groups (2020), Oberwolfach Reports, pp. 2739--2819, DOI 10.4171/owr/2019/44. https://doi.org/10.4171/owr/2019/44. This is the supplied workshop-report provenance; no exact problem page or individual attribution is supplied.
• Supplied literature source: Types for tame p-adic groups. https://doi.org/10.4007/annals.2021.193.1.4.
• Supplied literature source: Characteristic-free Yu construction. https://arxiv.org/abs/2605.03638.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the extracted statement as malformed and missing necessary objects and hypotheses, and reports generalizations of related Yu-type constructions in restricted settings. This is supplied dated curation, not an independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.