Invariant random subgroups on certain orbits
Manoj Choudhuri, C. R. E. Raja
TL;DR
The paper investigates invariant random subgroups (IRS) of a connected real Lie group $G$ that are supported on specific conjugation orbits inside the space ${ m Sub}_G$ of closed subgroups equipped with the $Chabauty$ topology. It develops a framework linking dynamics on ${ m Sub}_G$ with Grassmannian actions, proving a Furstenberg-type rigidity for unipotent dynamics and establishing when IRS can exist on Levi, maximal compact, Borel, and diagonalizable orbits. The main results show that a Levi IRS forces normality of the Levi subgroup, an IRS on maximal compact subgroups implies strong normality and structural decompositions, an IRS on Borel subgroups enforces amenability of $G$, and an IRS on maximal diagonalizable subgroups implies centrality of the diagonalizable part; converses provide explicit IRS in some cases. Together, these results tie IRS existence on these orbits to fundamental structural properties of $G$, offering a clear picture of when IRS can live on natural conjugation orbits and what that implies about the group's architecture.
Abstract
Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$.