Canonical Forms Under Certain Actions on the Classical Compact Simple Lie Groups
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.
KeywordsMaximal Torus Orbit Space Adjoint Action Outer Automorphism Holomorphic Isometry
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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