Orbit Decompositions in Wonderful Group Schemes

Definitions and mathematical statement

The supplied record does not specify the hypotheses on the field $k$, its valuation ring $\mathfrak{o}$, the group $G$, or the parameter $f$, nor does it give the construction of the object called the wonderful group scheme $G_{x,f}$. The following question is therefore conditional on recovering those data and hypotheses from the cited source.

Write $\mathcal{B}(G)$ for the Bruhat--Tits building intended in the source. Let $C\subseteq\mathcal{B}(G)$ be a chamber containing $x$, and put \[ I:=G(k)_{C,0},\qquad X:=G_{x,f}(\mathfrak{o}), \] where the source identifies $I$ as the associated Iwahori subgroup and $X$ denotes the set of $\mathfrak{o}$-valued points. The convention for chambers and whether boundary points are permitted are not specified.

Orbit-classification question. For the intended action of $I\times I$ on $X$, classify all orbits. More explicitly, after the action \[ \alpha:(I\times I)\times X\longrightarrow X \] has been specified, determine a parameter set $A$ and a complete invariant $q:X\to A$ such that $q$ is surjective and, for every $u,v\in X$, \[ q(u)=q(v) \quad\Longleftrightarrow\quad \exists(i_1,i_2)\in I\times I:\ v=\alpha((i_1,i_2),u). \] This is the requested analogue of the classification of $(B\times_k B)$-orbits in the wonderful compactification mentioned in the excerpt. Neither $B$ nor the action on $X$ is defined in the supplied material; in particular, a left--right multiplication formula is not imposed here.

Stable-piece question. Describe the diagonal-stable pieces of $X$ intended by the source, in analogy with the stable-piece decomposition of the wonderful compactification. For a group $H$, its diagonal subgroup is \[ \operatorname{Diag}(H):=\{(h,h):h\in H\}\subseteq H\times H. \] The original excerpt specifically refers to $G_{x,f}(\mathfrak{o})^{\operatorname{Diag}}$-stable pieces in $G_{x,f}(\mathfrak{o})$. It does not define the relevant diagonal action or the pieces. Thus no precise partition, parametrization, or equivalence with diagonal-subgroup orbits can be asserted from this record alone.

The excerpt motivates both questions by similarities with the wonderful compactification when $f=0$. It describes its Theorems 1 and 3 as possible first steps toward a Bruhat--Tits theory for wonderful compactifications, and expresses an expectation of extending the results to more general wonderful varieties in the sense of its reference [4]. These are source qualifications and motivations, not additional established conclusions or specified extension statements.

Short English statement

For a chamber $C$ containing $x$, classify the orbits of the associated Iwahori subgroup product $G(k)_{C,0}\times G(k)_{C,0}$ on $G_{x,f}(\mathfrak{o})$, and describe the intended diagonal-stable pieces, by analogy with the wonderful compactification. The construction, hypotheses, actions, and meaning of stable pieces require clarification from the source before this becomes a fully specified mathematical problem.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30006259$, title "Orbit Decompositions in Wonderful Group Schemes," exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2025, and names no individual proposer.
• Supplied report citation: Algebraic Groups (2025), Oberwolfach Reports, pp. 941--1010, DOI 10.4171/owr/2025/20. https://doi.org/10.4171/owr/2025/20.
• Supplied literature source, "OWR 2025 report": https://ems.press/content/serial-article-files/51849.
• Supplied literature source, "Li, wonderful embeddings for group schemes": https://arxiv.org/abs/2505.12777. No fuller author attribution or bibliographic details are supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that the wonderful group-scheme construction is available while the requested Iwahori orbit classification and diagonal-stable-piece description remain open questions. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked and notation repaired. This is supplied verification metadata, not verification performed for this draft. The excerpt refers to [2, § 2.1], [5], [4], and Theorems 1 and 3, but their bibliographic identities or statements are not included in the supplied record.