Agreement of Affine Deligne–Lusztig Stratifications

Definitions and mathematical statement

The supplied question concerns affine Deligne--Lusztig varieties and an invariant denoted by $f_g$, compared with the generalized Ekedahl--Oort invariant. The record does not define either invariant, specify their domains, or enumerate the cases studied by Görtz and He. Consequently, only the following conditional formulation can be made precise from the supplied information.

Let $\mathcal C_{\mathrm{GH}}$ denote the intended collection of cases studied by Görtz and He. For each $c\in\mathcal C_{\mathrm{GH}}$, let $X_c$ be the common underlying set on which the two invariants are to be compared, and write \[ F_c:X_c\longrightarrow A_c,\qquad E_c:X_c\longrightarrow B_c \] for their value maps. Here $F_c$ is a bookkeeping symbol for the invariant called $f_g$ in the source, and $E_c$ denotes the generalized Ekedahl--Oort invariant; these symbols do not supply the missing definitions. For a map $H:X\to Y$, define its fiber partition by \[ \mathcal P(H)=\{H^{-1}(y):y\in H(X)\}. \] The intended question, interpreted as equality of fiber partitions, is whether \[ \forall c\in\mathcal C_{\mathrm{GH}},\qquad \mathcal P(F_c)=\mathcal P(E_c). \] Equivalently, is it true that \[ \forall c\in\mathcal C_{\mathrm{GH}}\ \forall x,y\in X_c,\qquad F_c(x)=F_c(y)\ \Longleftrightarrow\ E_c(x)=E_c(y)? \] The cases, geometric data, admissibility hypotheses, meaning of the index $g$, and actual definitions of the maps require clarification from the cited sources. Equality of fiber partitions expresses only a set-theoretic interpretation of "the same stratification"; the record does not specify whether additional geometric structure must agree.

Short English statement

Does the invariant $f_g$ distinguish exactly the same strata as the generalized Ekedahl--Oort invariant in every case studied by Görtz and He? The supplied record leaves the cases and invariants undefined, so the displayed formulation is conditional. Its supplied status is partially solved; that status is not independently verified here.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30002949$, source problem code catalogue entry, title "Agreement of Affine Deligne--Lusztig Stratifications," in the Oberwolfach Reports Open Problems collection. No proposer is identified. The recorded proposed year is 2015.
• Supplied report citation: Reductions of Shimura Varieties (2016), Oberwolfach Reports, pp. 2265--2328, DOI 10.4171/owr/2015/39. Original OWR report: https://doi.org/10.4171/owr/2015/39. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, "full report text": https://ems.press/content/serial-article-files/46591.
• Supplied literature source, "Goertz He and Nie 2022": https://doi.org/10.4153/S0008414X22000608. No article title or more detailed bibliographic information is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the original work verifies the Vollaard--Wedhorn case and one additional case, while later Coxeter type work advances the classification but still records open cases. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the cleanup is meaning-preserving and self-contained. This is supplied editorial metadata, not independent mathematical verification of the formulation above.