Group-Level Realization of the c-Operation

Definitions and mathematical statement

Let $G$ be a reductive $p$-adic group in the scope intended by the source. Write $\mathcal H_I$ for its Iwahori--Hecke algebra and $c_{\mathcal H_I}$ for the operation called the $c$-operation in the supplied record. These are placeholders for source-specific constructions: the record does not define this operation or specify the coefficient ring, algebra conventions, or hypotheses on $G$.

Requested construction. Construct a realization of the $c$-operation for $G$ itself that recovers $c_{\mathcal H_I}$ on its Iwahori--Hecke algebra.

The supplied information does not specify what mathematical object constitutes a realization for $G$, or the comparison procedure by which it must recover the algebra-level operation. Consequently, this is a statement schema requiring clarification from the cited report, rather than a fully specified construction problem. In particular, no particular involution, representation-theoretic functor, or compatibility identity is asserted here. The supplied assessment mentions Bernstein projectives as a proposed route; it does not supply a construction or a theorem establishing the requested realization.

Short English statement

The problem asks for a realization of the $c$-operation for a reductive $p$-adic group that recovers the known Iwahori--Hecke algebra operation. The operation and the required comparison are undefined in the supplied record, so the target remains incomplete. The supplied open status requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004257$, title: Group-Level Realization of the c-Operation; source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and lists proposed year $2019$; no individual proposer is identified.
• 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 citation is supplied provenance; the report has not been independently consulted for this draft.
• Supplied literature source: OWR 2019 44. https://ems.press/content/serial-article-files/46824.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It labels the problem open, describes the group-level realization as a natural question, mentions Bernstein projectives as a proposed route, and reports that no general construction or resolving theorem was found in that curation. This does not establish the absence of a solution.
• The registry labels the statement corrected_verified and says that notation, typography, and sentence structure were repaired while preserving mathematical content. These are supplied editorial claims, not mathematical verification of this draft.