Cayley transform of a theta-stable Cartan subalgebra

Assume the Axiom of Choice (The Axiom of Choice). Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Killing form $B_0$ and Cartan involution $\theta$, let ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ be the Cartan decomposition (Cartan involution of a real semisimple Lie algebra, Cartan decomposition of a real semisimple Lie algebra), let $\mathfrak{g}={\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}$ be the complexification with its Killing form $B$, its conjugation $\sigma$, and the $\mathbb{C}$-linear extension of $\theta$, again written $\theta$, so that $\theta \sigma =\sigma \theta$ (Complexification of a real Lie algebra). Let ${\mathfrak{h}}_0$ be a $\theta$-stable Cartan subalgebra with ${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$ and complexification $\mathfrak{h}={\mathfrak{h}}_0\oplus i{\mathfrak{h}}_0$ (Theta-stable Cartan subalgebras and their compact and split parts), and let $\mathrm{\Phi }=\mathrm{\Phi }(\mathfrak{g},\mathfrak{h})$ be the root system of $(\mathfrak{g},\mathfrak{h})$ with root spaces ${\mathfrak{g}}_{\alpha }$ (Root and root space).

Roots of a $\theta$-stable Cartan subalgebra. For $T\in {\mathfrak{t}}_0$ the operator ${\mathrm{ad}}_T$ is skew-adjoint and for $A\in {\mathfrak{a}}_0$ it is self-adjoint for the positive definite form $B_{\theta }$ (Bracket relations and Killing signs in a Cartan decomposition), so every root $\alpha \in \mathrm{\Phi }$ satisfies $\alpha ({\mathfrak{t}}_0)\subseteq i\mathbb{R}$ and $\alpha ({\mathfrak{a}}_0)\subseteq \mathbb{R}$. By definition $\theta \alpha :=\alpha \circ {\theta }^{-1}$, and a root $\alpha$ is

$\text{real if }\alpha ({\mathfrak{t}}_0)=0,\text{imaginary if }\alpha ({\mathfrak{a}}_0)=0,\text{complex otherwise,}$

equivalently $\theta \alpha =-\alpha$, $\theta \alpha =\alpha$, or neither. If $\alpha$ is imaginary then $\theta ({\mathfrak{g}}_{\alpha })={\mathfrak{g}}_{\theta \alpha }={\mathfrak{g}}_{\alpha }$, so ${\mathfrak{g}}_{\alpha }$ is $\theta$-stable; being one-dimensional it satisfies ${\mathfrak{g}}_{\alpha }\subseteq \mathfrak{k}$ or ${\mathfrak{g}}_{\alpha }\subseteq \mathfrak{p}$, where $\mathfrak{k},\mathfrak{p}$ are the $\pm 1$-eigenspaces of $\theta$ in $\mathfrak{g}$. The imaginary root $\alpha$ is compact in the first case and noncompact in the second.

The normalizing norm. Write $\bar{Z}:=\sigma Z$ and let $B_{\theta }(Z,W):=-B(Z,\theta \bar{W})$ be the positive definite Hermitian form on $\mathfrak{g}$ extending the inner product $B_{\theta }$ on ${\mathfrak{g}}_0$ (Bracket relations and Killing signs in a Cartan decomposition). Let $H_{\alpha }\in \mathfrak{h}$ be the Killing-dual vector of $\alpha$ (Killing-dual vector of a root) and $h_{\alpha }=2H_{\alpha }\mathrm{/}\alpha (H_{\alpha })$ the coroot (Coroot of a Lie-algebra root). For a real root $H_{\alpha }\in {\mathfrak{a}}_0$ and for an imaginary root $H_{\alpha }\in i{\mathfrak{t}}_0$; in both cases

$\mathrm{\mid }\alpha {\mathrm{\mid }}^2:=B_{\theta }(H_{\alpha },H_{\alpha })=\mathrm{\mid }B(H_{\alpha },H_{\alpha })\mathrm{\mid }=\mathrm{\mid }\alpha (H_{\alpha })\mathrm{\mid }>0.$

The two Cayley transforms. Let $\alpha \in \mathrm{\Phi }$.

1. Noncompact imaginary root. Let $\beta$ be a noncompact imaginary root, so that ${\mathfrak{g}}_{\beta }\subseteq \mathfrak{p}$, and choose a nonzero $E_{\beta }\in {\mathfrak{g}}_{\beta }$ normalized by

$B(E_{\beta },\sigma E_{\beta })=\frac{2}{\mathrm{\mid }\beta {\mathrm{\mid }}^2}\mathrm{.}$

Such a choice exists: $B(E,\sigma E)=\frac{1}{2}B(E+\sigma E,E+\sigma E)>0$ for $0\mathrm{\ne }E\in {\mathfrak{g}}_{\beta }$ because $E+\sigma E\in {\mathfrak{p}}_0\setminus \{0\}$, and real rescalings $E\mapsto tE$, $t\in {\mathbb{R}}^{\times }$, scale $B(E,\sigma E)$ by $t^2$. The noncompact-imaginary Cayley transform of ${\mathfrak{h}}_0$ attached to $E_{\beta }$ is the complex automorphism

$c_{\beta }:=\exp (\frac{\pi }{4}\mathrm{ad}(\sigma E_{\beta }-E_{\beta }))\in \mathrm{Aut}(\mathfrak{g}),$

and the associated subspace is $c_{\beta }⟨{\mathfrak{h}}_0⟩:={\mathfrak{g}}_0\cap c_{\beta }(\mathfrak{h})$.

2. Real root. Let $\alpha$ be a real root, so that ${\mathfrak{g}}_{\alpha }$ is $\sigma$-stable, and choose a nonzero $E_{\alpha }\in {\mathfrak{g}}_{\alpha }\cap {\mathfrak{g}}_0$ normalized by

$B(E_{\alpha },\theta E_{\alpha })=-\frac{2}{\mathrm{\mid }\alpha {\mathrm{\mid }}^2}\mathrm{.}$

Such a choice exists: ${\mathfrak{g}}_{\alpha }\cap {\mathfrak{g}}_0\mathrm{\ne }0$ because $\theta \alpha =-\alpha$ gives $\sigma ({\mathfrak{g}}_{\alpha })={\mathfrak{g}}_{\alpha }$, $B(E,\theta E)=\frac{1}{2}B(E+\theta E,E+\theta E)<0$ for $0\mathrm{\ne }E\in {\mathfrak{g}}_{\alpha }\cap {\mathfrak{g}}_0$ because $E+\theta E\in {\mathfrak{k}}_0\setminus \{0\}$, and real rescalings scale $B(E,\theta E)$ by $t^2$. The real-root Cayley transform of ${\mathfrak{h}}_0$ attached to $E_{\alpha }$ is the complex automorphism

$d_{\alpha }:=\exp (i\frac{\pi }{4}\mathrm{ad}(\theta E_{\alpha }-E_{\alpha }))\in \mathrm{Aut}(\mathfrak{g}),$

and the associated subspace is $d_{\alpha }⟨{\mathfrak{h}}_0⟩:={\mathfrak{g}}_0\cap d_{\alpha }(\mathfrak{h})$.

In both cases the normalizations are exactly the ones that make $(E,\sigma E,h_{\beta })$ for an imaginary root and $(E,-\theta E,h_{\alpha })$ for a real root a root ${\mathfrak{s}\mathfrak{l}}_2$-triple: with the identity $[X,Y]=B(X,Y)H_{\alpha }$ for $X\in {\mathfrak{g}}_{\alpha }$, $Y\in {\mathfrak{g}}_{-\alpha }$ (which follows from invariance of $B$, the nondegeneracy of $B{\mathrm{\mid }}_{\mathfrak{h}}$ and $\alpha (H)=B(H_{\alpha },H)$; Trace forms are symmetric and invariant, Killing form) one gets $[E_{\beta },\sigma E_{\beta }]=h_{\beta }$ and $[E_{\alpha },-\theta E_{\alpha }]=h_{\alpha }$.

Dependence on the choices. The transforms depend on the chosen normalized root vectors $E_{\alpha }$, and the associated subspaces as well as the two kinds of transforms are the objects used in Cayley transforms connect theta-stable Cartans in the classification, where their geometrical effect is computed for an arbitrary such choice. The real-root construction and the noncompact-imaginary construction are inverse to one another in the precise sense recorded there.