Finite type multiple flag varieties of exceptional groups

TL;DR

This paper develops an algorithm to determine when a simple complex Lie group acts with finitely many orbits on triple flag varieties, providing a complete classification for type F4 groups.

Contribution

It introduces a systematic algorithm for classifying finite orbit actions on triple flag varieties and applies it to classify type F4 cases.

Findings

Classified finite orbit actions for type F4 groups.

Developed an algorithm applicable to other types.

Lays groundwork for future classifications of E6, E7, E8.

Abstract

Consider a simple complex Lie group $G$ acting diagonally on a triple flag variety $G/P_1\times G/P_2\times G/P_3$, where $P_i$ is parabolic subgroup of $G$. We provide an algorithm for systematically checking when this action has finitely many orbits. We then use this method to give a complete classification for when $G$ is of type $F_4$. The $E_6, E_7,$ and $E_8$ cases will be treated in a subsequent paper.