Spherical homogeneous spaces of minimal rank

TL;DR

This paper classifies spherical pairs of minimal rank in complex reductive algebraic groups, revealing their structure and properties, which generalize well-known classes like tori and complete homogeneous spaces.

Contribution

It provides a classification of spherical pairs of minimal rank, expanding understanding of their structure within complex reductive algebraic groups.

Findings

Classification of spherical pairs of minimal rank

Identification of properties of these pairs

Extension of known classes like tori and G

Abstract

Let $G$ be a complex connected reductive algebraic group and $G/B$ denote the flag variety of $G$. A $G$-homogeneous space $G/H$ is said to be {\it spherical} if $H$ acts on $G/B$ with finitely many orbits. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group $G$ (viewed as a $G\times G$-homogeneous space) has particularly nice proterties. Namely, the pair $(G,H)$ is called a {\it spherical pair of minimal rank} if there exists $x$ in $G/B$ such that the orbit $H.x$ of $x$ by $H$ is open in $G/B$ and the stabilizer $H_x$ of $x$ in $H$ contains a maximal torus of $H$. In this article, we study and classify the spherical pairs of minimal rank.