Realizations of globally exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$- symmetric spaces

TL;DR

This paper explicitly constructs and classifies the globally realized exceptional $ ext{Z}_2 imes ext{Z}_2$-symmetric spaces associated with exceptional compact Lie groups, expanding on prior classification results.

Contribution

It provides concrete pairs of commuting involutions and determines the structure of their fixed point groups, realizing the symmetric spaces globally.

Findings

Explicit pairs of involutions for exceptional groups

Determination of fixed point group structures

Global realization of symmetric spaces

Abstract

A classification is given of the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric spaces $G/K$ by A.Kollross, where $G$ is an exceptional compact Lie group or $S\!pin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms (involutions) $\tilde{\sigma}, \tilde{\tau}$ of $G$ concretely and determine the structure of group $G^{\sigma} \cap G^{\tau}$ corresponding to Lie algebra $\mathfrak{g}^\sigma \cap \mathfrak{g}^\tau$, where $G$ is an exceptional compact Lie group. Thereby, we realize exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric spaces, globally.