Moment Polyhedra of Noncompact Coadjoint Orbits

Definitions and mathematical statement

Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$, let $K\subseteq G$ be a compact subgroup with Lie algebra $\mathfrak k$, and let $O=\operatorname{Ad}^{*}(G)\lambda\subseteq\mathfrak g^{*}$ be a coadjoint orbit, where $\lambda\in\mathfrak g^{*}$. The orbit projection is the restriction map \[ \mu_K:O\longrightarrow\mathfrak k^{*},\qquad \mu_K(\xi)=\xi|_{\mathfrak k}. \] With the compatible sign convention for the Kirillov--Kostant--Souriau symplectic form, this is a moment map for the restricted coadjoint action of $K$.

To describe the usual chamber construction, let $K^{\circ}$ be the identity component of $K$, choose a maximal torus $T\subseteq K^{\circ}$, and choose a closed positive Weyl chamber $\mathfrak t^{*}_{+}$. A $K$-invariant inner product on $\mathfrak k$ allows this chamber to be regarded as a subset of $\mathfrak k^{*}$. Define the chamber image and its closure by \[ S_K(O)=\mu_K(O)\cap\mathfrak t^{*}_{+},\qquad \overline{S_K(O)}=\operatorname{cl}_{\mathfrak t^{*}_{+}}S_K(O). \] These definitions do not assert that either set is convex or polyhedral.

Source problem. Given such $G$, $K$, and $O$, determine explicit equations describing the moment polyhedron $\Delta_K(O)$ of the orbit projection $\mu_K$.

Unresolved qualifications. The supplied statement does not define $\Delta_K(O)$, specify whether it denotes the chamber image or its closure, or supply hypotheses guaranteeing convexity or polyhedrality. It also does not specify whether "equations" includes affine inequalities or requires a finite description. The chamber construction above uses $K^{\circ}$; if disconnected compact subgroups are intended, the treatment of their additional component action must be specified. No compactness of $G$ or $O$, properness of $\mu_K$, reductivity, or holomorphic-orbit hypothesis is imposed in the supplied statement. Accordingly, the requested description is recorded in its supplied generality, with these conventions and hypotheses unresolved.

Short English statement

For a coadjoint orbit of a connected Lie group, describe explicitly the moment image obtained by restricting covectors to the Lie algebra of a compact subgroup and passing to a positive Weyl chamber. The source calls the target a moment polyhedron, but leaves its closure convention and the hypotheses ensuring polyhedrality unspecified. Its partially solved label is supplied dated curation and requires independent verification.

Sources

• Supplied local problem registry record, permanent UnsolvedMath ID $30001706$, title Moment Polyhedra of Noncompact Coadjoint Orbits, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2011 and names no individual proposer.
• Supplied report citation: Geometric Quantization in the Non-compact Setting (2011), Oberwolfach Reports, pp. 425--521, DOI 10.4171/owr/2011/09. Original report: https://doi.org/10.4171/owr/2011/09. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Deltour, Kirwan polyhedron of holomorphic coadjoint orbits, https://arxiv.org/abs/1104.1909.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that Berenstein--Sjamaar solve the compact case, while Deltour gives affine equations for holomorphic noncompact coadjoint orbits; it states that the full extracted generality remains broader. This is supplied dated curation, not an independently verified literature assessment. No separate bibliographic reference for the Berenstein--Sjamaar claim is supplied.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of its cleanup. These are supplied editorial assertions, not mathematical verification of this draft or independent inspection of the cited sources.