Orbit types of the compact Lie group E_7 in the complex Freudenthal vector space P^C

TL;DR

This paper classifies the orbit types of the compact Lie group E_7 acting on the complex Freudenthal vector space, extending known classifications for smaller exceptional groups.

Contribution

It determines the seven orbit types of E_7 in the complex Freudenthal vector space, a novel classification for this group in this context.

Findings

E_7 has seven orbit types in P^C.

Orbit types include E_7/E_7, E_7/E_6, E_7/F_4, E_7/Spin(11), E_7/Spin(10), E_7/Spin(9), E_7/Spin(8).

Extends classification of orbit types from smaller exceptional groups.

Abstract

Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbit types which are E_6/E_6, E_6/F_4, E_6/Spin(10), E_6/Spin(9), E_6/Spin(8) . In this paper, we determine the orbit types of the simply connected compact exceptional Lie group E_7 in the complex Freudenthal vector space P^C. As results, E_7 has seven orbit types which are E_7/E_7, E_7/E_6, E_7/F_4, E_7/Spin(11), E_7/Spin(10), E_7/Spin(9), E_7/Spin(8).