I am trying to learn the basics of Cartan decomposition of Lie algebra, and have come across the following example.
Consider $\mathfrak{gl_n}$ as the Lie algebra of endomorphisms of $\mathbb{C}^n$. Let the Cartan involution on $\mathfrak{gl_n}$ be given by $\theta (A)=-A^*, $ where $*$ denotes conjugate transpose of $A \in \mathfrak{gl_n}$ and the killing form is given by $$(X,Y)=-\frac{1}{2} Tr(A\theta(B)).$$ I have the following questions:
$1)$ What does it mean by $\pm1-$eigenspace of $\theta$ ? This wikipedia page tells that
Since $\theta ^ 2 =1,$ the linear map $\theta$ has the two eigenvalues $\pm1.$
What is the definition of eigen-space in this context? I know eigenspaces only in the context of basic linear algebra. I got confused here.
$2)$ What is the Cartan decomposition of $\mathfrak{gl}_n$ ? Any suggestions for references?
Thanks!
The map $\theta$ is a linear map of the vector space $\mathfrak{gl}_n$. Eigenvectors of $\theta$ are matrices $X$ such that $-X^* = \lambda X$. Since $\theta^2$ is the identity function, eigenvalues of $\theta$ must be square roots of $1$, namely $\pm 1$.
The eigenspace corresponding to the eigenvalue $1$ is the set $$ \left\{X \in \mathfrak{gl_n} : -X^* =X\right\} $$ This is the set of skew-hermitian matrices. The $-1$-eigenspace is the set $$ \left\{X \in \mathfrak{gl_n} : X^* =X\right\} $$ of hermitian matrices.
Look in Helgason's book Differential Geometry, Lie Groups, and Symmetric Spaces for more.