Survey of Smooth Modular Representations

Definitions and mathematical statement

Formulation status. The supplied record identifies an open-ended survey entitled Survey of Smooth Modular Representations, rather than a recoverable standalone conjecture. Its statement status is unrecoverable. Consequently, no precise open mathematical assertion, with complete hypotheses and quantifiers, can be extracted from this record alone.

Recoverable notation. Whenever a group $G$, a coefficient ring $R$, a subgroup $K$, and a category $\operatorname{Mod}_R(G)$ of smooth representations have been specified, write \[ V^K=\{v\in V:kv=v\text{ for every }k\in K\}. \] The condition that $V$ is generated by its $K$-invariant vectors means \[ V=\operatorname{span}_R\{gv:g\in G,\ v\in V^K\}. \] Let $\mathcal C_K$ denote the full subcategory of such representations, and let $\operatorname{Irr}_R(G)$ denote the isomorphism classes of irreducible objects of $\operatorname{Mod}_R(G)$. These definitions explain the extracted notation; they do not determine the missing hypotheses on $G$, $R$, or $K$.

Extracted assertions requiring source recovery. The fragment denotes an induced representation by $P=\operatorname{ind}_K^G(1_R)$, with $1_R$ apparently the trivial $K$-representation on $R$. It reports that, if $P$ is projective, the invariants assignment $V\mapsto V^K$ gives a bijection of the schematic form \[ \{[V]\in\operatorname{Irr}_R(G):V^K\ne0\} \longleftrightarrow \{[M]:M\text{ is a simple }H_R(G,K)\text{-module}\}, \] and an equivalence \[ \mathcal C_K\simeq\operatorname{Mod}(H_R(G,K)). \] The fragment also mentions stability under subrepresentations, an extension from projectivity to "almost projectivity," and an Iwahori subgroup as an example. These are reported source claims, not assertions validated by this draft: the induction convention, coefficient hypotheses, definition of almost projectivity, Hecke algebra definition, module handedness, and precise role of stability under subrepresentations are absent or unclear. The opening claim about a convolution algebra being isomorphic to an opposite algebra has lost an essential referent.

A detached footnote appears to qualify some assertion as true for mod $p$ representations of $\operatorname{GL}_2(\mathbb Q_p)$ but false in general. Both the attachment of the footnote and the restoration of its damaged group notation require verification; no assertion above is assigned that qualification without evidence.

Unresolved survey target. The recoverable question is only what is known and unknown about smooth modular representations and their Hecke algebras. The supplied material does not identify a particular classification problem, universal assertion, or conjectural equivalence to formalize as an open problem.

Short English statement

This record contains a survey heading and a damaged passage about invariants and Hecke algebras. It does not supply a standalone mathematical problem. The displayed correspondences describe the extracted claims schematically; missing hypotheses, an undefined almost-projectivity condition, and a detached qualification prevent a precise theorem or conjecture statement. The supplied open label requires independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied complete record for permanent UnsolvedMath ID $30006115$, exact source code catalogue entry, title Survey of Smooth Modular Representations. No individual proposer is supplied; the proposed year is $2024$.
• 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. This report citation is supplied provenance; the relevant passage has not been independently inspected for this draft.
• Supplied literature source, labeled "Vigneras survey report": https://publications.mfo.de/bitstream/handle/mfo/4270/OWR_2024_54.pdf?sequence=1. The record attributes the material to an Oberwolfach workshop survey and its open-problem discussions; it supplies no more precise passage location or individual problem attribution.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. The supplied assessment labels the record open while explaining that the extracted text is an open-ended survey prompt rather than a resolved mathematical assertion. The supplied statement-verification note likewise identifies "what is known and what is not known?" as a broad survey heading, not a standalone problem. The record requests independent verification of time-sensitive status and cited claims.