On the orbit space of a maximal compact subgroup on a spherical homogeneous variety

TL;DR

This paper establishes a homeomorphism between the orbit space of a spherical homogeneous variety under a maximal compact subgroup and its valuation cone, linking geometric and combinatorial structures.

Contribution

It proves the orbit space is homeomorphic to the valuation cone and explores the relationship between orbit type stratification and face stratification.

Findings

Orbit space is homeomorphic to the valuation cone.

Relation between orbit type stratification and face stratification.

Provides a geometric interpretation of valuation cones.

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.