A restriction on proper actions on homogeneous spaces of reductive type

TL;DR

This paper establishes a necessary condition for the proper action of reductive subgroups on homogeneous spaces of reductive type, leading to examples where standard compact Clifford-Klein forms cannot exist.

Contribution

It introduces a new necessary condition for proper actions on reductive homogeneous spaces and applies it to identify spaces lacking standard compact Clifford-Klein forms.

Findings

Identified a necessary condition for proper actions of reductive subgroups.

Provided examples of spaces without standard compact Clifford-Klein forms.

Abstract

Let L be a reductive subgroup of a reductive Lie group G. Let G/H be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of L on G/H. As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.