On the orbit space of a maximal compact subgroup on a spherical homogeneous variety
Dmitry A. Timashev
TL;DR
The paper addresses the topological structure of the orbit space $X/K$ for a spherical homogeneous variety $X=G/H$ under a maximal compact subgroup $K$, proving the homeomorphism $X/K \cong \mathcal{V}_X$. The authors employ a smooth projective toroidal compactification $\overline{X}$ and its invariant moment polytope $\mathcal{P}_{\overline{X}}$, together with the moment map $\Phi$ and Kirwan-type map $\Psi$, to connect orbit types with the face structure of the polytope via BLV-slices and satellite subgroups. They show that the orbit-type stratification on $X/K$ corresponds to the face stratification of $\mathcal{P}_{\overline{X}}$, while horospherical cases yield the full cone $\mathcal{V}_X$, and in general the stratifications can differ in refinement. The work thus provides a symplectic- and convex-geometry-based framework for understanding degenerations of $H$ and the $K$-orbit structure, with concrete examples illustrating both the reach and the limitations of the correspondence.
Abstract
Let $X=G/H$ be a spherical homogeneous variety for a complex reductive algebraic group $G$. We prove that the orbit space of $X$ under the action of a maximal compact subgroup $K\subset G$ is homeomorphic to the valuation cone of $X$. We also discuss the relation between the orbit type stratification of the orbit space and the face stratification of the valuation cone.