Spaces defined by Lie group endomorphisms ($\Phi$-spaces)

A. S. Fedenko

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 4 (1973), pp. 231-267

Résumé

А $\Phi$-space is a triple $(G,H,\Phi)$ consisting of a connected Lie group $G$, a closed subgroup $H$ of $G$ and an analytic endomorphism $\Phi$ of $G$ such that $H$ lies between $G^\Phi$ and the identity component of $G^\Phi$, where $G^\Phi$ denotes the closed subgroup of $G$ formed by all the elements left fixed by $\Phi$. The symmetric space is obtained in the case $\Phi^2=1$. A local triple is triple $\mathfrak g,\mathfrak h,\varphi$ consisting of the finite-dimensional real Lie algebra $\mathfrak g$, a subalgebra $\mathfrak h$ of $\mathfrak g$, and an endomorphism $\varphi$ of $\mathfrak g$ such that $\mathfrak h$ consist of all elements of $\mathfrak g$ left fixed by $\varphi$. If $\varphi^2=1$, then $(\mathfrak g,\mathfrak h,\varphi)$ is a symmetric Lie algebra. Every $\Phi$-space $(G,H,\Phi)$ gives rise to a local triple $(\mathfrak g,\mathfrak h,\varphi)$ in a natural manner; $\mathfrak g$ and $\mathfrak h$ are the Lie algebras of $G$ and $H$, respectively, and $\varphi=d\Phi_e$, where $e$ is identity element of $G$. Conversely, if $(\mathfrak g,\mathfrak h,\varphi)$ is a local triple and if $G$ is a connected, simply connected Lie group with Lie algebra $\mathfrak g$, then the endomorphism $\varphi$ of $\mathfrak g$ induces an endomorphism $\Phi$ of $G$ and, for any subgroup $H$ lying between $G^\Phi$ and the identity component of $G^\Phi$, the triple $(G,H,\Phi)$ is а $\Phi$-space. In § 4 all $\Phi$-spaces are described which generate the local triple $(\mathfrak g,\mathfrak h,\varphi)$, where $\mathfrak g$ is the semi-simple compact Lie algebra. In § 5 $\Phi$-space $(G,Н,\Phi)$ is considered for which $G$ is the semisimple compact Lie group acting effectively on $G/H$ and $\Phi$ is of finite order. It is shown that the identity component of a group of isometries of space $G/H$ coincides with $G$. In § 6 a classification of periodic automorphisms of classical compact groups and corresponding $\Phi$-spaces is given. In § 7 is considered a corresponding classification for classical noncompact groups. This classification is based on the results of § 6 and on a Theorem in [20]. Let $G$ be a Lie group with a finite number of connected compo nents, and let $A$ be a finite subgroup of the group of automorphisms of $G$. Then there exists a maximal ompact subgroup $K$ of $G$ which is stable under $A$.