Orbit Structure of Maximal Flat Flag Degenerations

Definitions and mathematical statement

The supplied record names a group $G=\operatorname{Aut}_{Q}(M)$, a space $X=\operatorname{Fl}^{mf}(V)$, and a natural action $G\curvearrowright X$. It does not define $Q$, $M$, $V$, the superscript $mf$, or the action. These symbols are therefore retained as source notation, without assigning unsupported hypotheses.

For a specified action, the orbit of $x\in X$ and the orbit set are \[ G\cdot x=\{g\cdot x:g\in G\},\qquad G\backslash X=\{G\cdot x:x\in X\}. \] The requested problem is to determine whether \[ \lvert G\backslash X\rvert<\infty, \] to give an explicit parametrization $\lambda\mapsto\mathcal O_\lambda$ that is a bijection from a described parameter set $\Lambda$ onto $G\backslash X$, and to determine exactly which pairs $\lambda,\mu\in\Lambda$ satisfy \[ \mathcal O_\lambda\subseteq\overline{\mathcal O_\mu}^{\,X}. \] Here the closure must be taken in the intended topology on $X$, which the supplied record does not specify. The underlying field, permitted objects, flag conditions, and scope of quantification over $Q,M,V$ are also absent. Consequently, this is a precise description of the requested orbit questions conditional on the missing setup, rather than a fully specified mathematical problem.

Short English statement

For the action denoted by $\operatorname{Aut}_{Q}(M)\curvearrowright\operatorname{Fl}^{mf}(V)$, determine whether there are finitely many orbits, describe parameters distinguishing them, and characterize inclusion of one orbit in the closure of another. The supplied record omits the definitions and hypotheses needed to specify the action, flags, and closure topology.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004267$, exact source problem code catalogue entry, titled "Orbit Structure of Maximal Flat Flag Degenerations." No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, pp. 2869--2909, DOI 10.4171/owr/2019/46. https://doi.org/10.4171/owr/2019/46.
• Supplied literature source, labeled "OWR 2019 46": https://ems.press/content/serial-article-files/46825.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the report explicitly requests the closure relation for a proposed stratification, and reports that no general closure theorem resolving it was found. This is supplied dated curation, not an independently verified literature assessment or a proof of current openness.
• The supplied record labels its statement corrected_verified and reports manual semantic, typographic, and LaTeX review asserting self-containment. That assertion is preserved as source metadata; the missing mathematical setup remains unresolved in this draft. Neither linked source has been independently consulted here.