Information outside the Thin Affine Grassmannian

Definitions and mathematical statement

Let $\operatorname{Gr}$ denote the affine Grassmannian associated with the group $\check G$ intended in the supplied report, and write $K=\check G({\mathbb{C}}[[z]])$, where ${\mathbb{C}}[[z]]$ is the ring of formal power series. The extract does not specify $\check G$ or its hypotheses. Write $\operatorname{Gr}^{\varpi}$ for the fixed-point locus of the action denoted by $\varpi$, and $\operatorname{Gr}_{\mathrm{thin}}$ for its designated thin part. Neither this action nor the thin part is defined in the supplied record.

1. Determine what representation-theoretic or geometric information is carried by the components of $\operatorname{Gr}^{\varpi}$ described in the source as lying outside $\operatorname{Gr}_{\mathrm{thin}}$.
2. Determine the relationship between Smith--Treumann localization for $K$-equivariant sheaves and Smith--Treumann localization for Iwahori--Whittaker sheaves, and compare the constructions and results attributed to Leslie--Lonergan and Riche--Williamson.

These are research questions whose precise targets require additional source definitions. In particular, the extract does not specify the meaning of component, whether the thin part is a union of components, the sheaf coefficients, the relevant categories, the localization functors, or the Iwahori--Whittaker datum. Thus no particular equivalence, commutative diagram, or classification is asserted here.

The supplied exposition attributes to Riche--Williamson the observation that Smith--Treumann localization and Smith parity sheaves also make sense for Iwahori--Whittaker sheaves, referring to its internal references [6] and [1]. It also attributes to [6] a short proof of a tilting character formula described as valid for all $p$; the formula and the domain of $p$ are not supplied. A detached sentence mentions a correspondence under geometric Satake with the Frobenius contraction functor of Gros--Kaneda, internal reference [2], but its antecedent is missing. These contextual assertions are preserved as source attributions, not as independently verified mathematical claims.

Short English statement

The problem asks what the components outside the thin part of a fixed-point affine Grassmannian encode, and how equivariant and Iwahori--Whittaker Smith--Treumann localization compare, including the Leslie--Lonergan and Riche--Williamson constructions. Essential definitions are absent from the extract. Supplied dated curation reports a later comparison of the localization constructions while retaining the outside-components question as unresolved; this assessment requires independent verification.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30006266$, exact source code catalogue entry, title Information outside the Thin Affine Grassmannian, in the Oberwolfach Reports Open Problems collection. The record assigns the year 2025 and supplies no individual proposer. Its labels open and corrected_verified, including its claim that the official report was checked and the notation repaired, are supplied registry metadata rather than verification performed for this draft.
• Enveloping Algebras and Geometric Representation Theory (2025), Oberwolfach Reports, pp. 1271--1332, DOI 10.4171/owr/2025/25. Supplied report citation: https://doi.org/10.4171/owr/2025/25. Supplied literature-source link, labeled "OWR 2025 report": https://ems.press/content/serial-article-files/51851.
• Supplied literature-source entry: Achar--Sobaje, Steinberg quotients and localization, https://arxiv.org/abs/2606.21697. The supplied literature assessment states that a later comparison relates the two Smith--Treumann localization constructions, while the question concerning information outside the thin affine Grassmannian remains unresolved.
• Literature assessment date: 2026-08-22. This is supplied dated curation, described in the record as literature triage derived from the OWR curation; its time-sensitive status and cited claims have not been independently verified here.
• The original extract names Leslie--Lonergan, Riche--Williamson, and Gros--Kaneda, and uses internal reference labels [1], [2], and [6]. The corresponding bibliography entries are not supplied, so these labels cannot be resolved reliably from this record alone.