Real and Complex Submanifolds pp 329-338 | Cite as

# Canonical Forms Under Certain Actions on the Classical Compact Simple Lie Groups

## Abstract

A maximal torus of a compact connected Lie group can be seen as a canonical form of adjoint action since any two maximal tori can be transformed each other by an inner automorphism. A. Kollross defined a *σ*-action on a compact Lie group which is a generalization of the adjoint action. Since a *σ*-action is hyperpolar, it has a canonical form called a section. In this paper we study the structure of the orbit space of a *σ*-action and properties of each orbit, such as minimal, austere and totally geodesic, using symmetric triads introduced by the author, when *σ* is an involution of outer type on the compact simple Lie groups of classical type. As an application, we investigate the fixed point set of a holomorphic isometry of an irreducible Hermitian symmetric space of compact type which does not belong to the identity component of the group of holomorphic isometries.

## Keywords

Maximal Torus Orbit Space Adjoint Action Outer Automorphism Holomorphic Isometry## Notes

### Acknowledgements

The author was partly supported by the Grant-in-Aid for Science Research (C) 2013 (No. 25400070), Japan Society for the Promotion of Science.

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