Decomposition Theorems for Triple Spaces

TL;DR

This paper investigates the geometric structure of triple spaces formed by specific Lie groups, establishing conditions for polar decompositions and orbit openness related to minimal parabolic subgroups.

Contribution

It provides new decomposition theorems for triple spaces involving groups like SL(2,R), SL(2,C), and SO_e(n,1), detailing conditions for polar decompositions and orbit structures.

Findings

Identified abelian subgroups allowing polar decomposition in triple spaces.

Determined conditions for minimal parabolic subgroups with open orbits.

Extended geometric understanding of triple spaces for specific Lie groups.

Abstract

A triple space is a homogeneous space $G/H$ where $G=G_0\times G_0\times G_0$ is a threefold product group and $H\simeq G_0$ the diagonal subgroup of $G$. This paper concerns the geometry of the triple spaces with $G_0=\SL(2,\R)$, $\SL(2,\C)$ or $\SO_e(n,1)$ for $n\ge 2$. We determine the abelian subgroups $A\subset G$ for which there is a polar decomposition $G=KAH$, and we determine for which minimal parabolic subgroups $P\subset G$, the orbit $PH$ is open in $G/H$.