Theta-stable Cartan subalgebras and their compact and split parts

Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Cartan involution $\theta$ and Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ (Cartan involution of a real semisimple Lie algebra, Cartan decomposition of a real semisimple Lie algebra). A Cartan subalgebra ${\mathfrak{h}}_0\subseteq {\mathfrak{g}}_0$ (Cartan subalgebra) is $\theta$-stable if $\theta ({\mathfrak{h}}_0)\subseteq {\mathfrak{h}}_0$; since $\theta$ is an involution, this is equivalent to $\theta ({\mathfrak{h}}_0)={\mathfrak{h}}_0$.

Let ${\mathfrak{h}}_0$ be a $\theta$-stable Cartan subalgebra. The restriction of $\theta$ to ${\mathfrak{h}}_0$ is an involutive linear map of ${\mathfrak{h}}_0$ preserving the bracket, so the two subspaces

${\mathfrak{t}}_0:={\mathfrak{h}}_0\cap {\mathfrak{k}}_0=\{X\in {\mathfrak{h}}_0:\theta X=X\},{\mathfrak{a}}_0:={\mathfrak{h}}_0\cap {\mathfrak{p}}_0=\{X\in {\mathfrak{h}}_0:\theta X=-X\}$

are the $+1$- and $-1$-eigenspaces of $\theta {\mathrm{\mid }}_{{\mathfrak{h}}_0}$ and

${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$

because every $X\in {\mathfrak{h}}_0$ decomposes as $X=\frac{1}{2}(X+\theta X)+\frac{1}{2}(X-\theta X)$ with both summands in ${\mathfrak{h}}_0$. Here ${\mathfrak{t}}_0$ is the compact part and ${\mathfrak{a}}_0$ the split part of ${\mathfrak{h}}_0$; the dimensions ${\dim }_{\mathbb{R}}{\mathfrak{t}}_0$ and ${\dim }_{\mathbb{R}}{\mathfrak{a}}_0$ are the compact dimension and the noncompact dimension of ${\mathfrak{h}}_0$. The Cartan subalgebra ${\mathfrak{h}}_0$ is maximally compact if its compact dimension is maximal among the $\theta$-stable Cartan subalgebras of ${\mathfrak{g}}_0$, and maximally noncompact, or maximally split, if its noncompact dimension is maximal. Whenever the collection of $\theta$-stable Cartan subalgebras is nonempty, both maxima exist because $0\le {\dim }_{\mathbb{R}}{\mathfrak{t}}_0\le {\dim }_{\mathbb{R}}{\mathfrak{k}}_0$ and $0\le {\dim }_{\mathbb{R}}{\mathfrak{a}}_0\le {\dim }_{\mathbb{R}}{\mathfrak{p}}_0$ for every $\theta$-stable Cartan subalgebra, the two dimensions being nonnegative integers bounded by the fixed dimensions of the summands of the Cartan decomposition. In particular, once a $\theta$-stable Cartan subalgebra has been specified, the collection is nonempty and each of these bounded sets of integer dimensions has a largest member. This conditional attainment argument does not assert the existence of a Cartan subalgebra.

The decomposition is compatible with the bracket in the following sense. Since $[{\mathfrak{k}}_0,{\mathfrak{k}}_0]\subseteq {\mathfrak{k}}_0$, $[{\mathfrak{k}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{p}}_0$ and $[{\mathfrak{p}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{k}}_0$ (Bracket relations and Killing signs in a Cartan decomposition) and since ${\mathfrak{h}}_0$ is closed under brackets, one has

$[{\mathfrak{t}}_0,{\mathfrak{t}}_0]\subseteq {\mathfrak{t}}_0,[{\mathfrak{t}}_0,{\mathfrak{a}}_0]\subseteq {\mathfrak{a}}_0,[{\mathfrak{a}}_0,{\mathfrak{a}}_0]\subseteq {\mathfrak{t}}_0\mathrm{.}$

Moreover the two summands are orthogonal for the Killing form $B$, and the restriction of $B$ is negative definite on ${\mathfrak{t}}_0$ and positive definite on ${\mathfrak{a}}_0$ (Bracket relations and Killing signs in a Cartan decomposition), so $B_{\theta }(X,Y)=-B(X,\theta Y)$ is a positive definite inner product on ${\mathfrak{h}}_0$ for which ${\mathfrak{t}}_0$ and ${\mathfrak{a}}_0$ are orthogonal.

The integers ${\dim }_{\mathbb{R}}{\mathfrak{t}}_0$ and ${\dim }_{\mathbb{R}}{\mathfrak{a}}_0$ are invariants of the conjugacy class of ${\mathfrak{h}}_0$. Indeed, let an automorphism $\alpha$ of ${\mathfrak{g}}_0$ carry ${\mathfrak{h}}_0$ onto another $\theta$-stable Cartan subalgebra ${\mathfrak{h}}_0^{\mathrm{\prime }}={\mathfrak{t}}_0^{\mathrm{\prime }}\oplus {\mathfrak{a}}_0^{\mathrm{\prime }}$. Every Lie-algebra automorphism preserves the Killing form, so $\alpha {\mathrm{\mid }}_{{\mathfrak{h}}_0}$ is an isometry from $B{\mathrm{\mid }}_{{\mathfrak{h}}_0}$ to $B{\mathrm{\mid }}_{{\mathfrak{h}}_0^{\mathrm{\prime }}}$. The displayed orthogonal decompositions show that the negative and positive inertia indices of these two restrictions are respectively $(\dim {\mathfrak{t}}_0,\dim {\mathfrak{a}}_0)$ and $(\dim {\mathfrak{t}}_0^{\mathrm{\prime }},\dim {\mathfrak{a}}_0^{\mathrm{\prime }})$. Sylvester's law of inertia (Sylvester's law of inertia: every real symmetric form is congruent to $\mathrm{diag}(I_p,-I_q,0_r)$, and $(p,q,r)$ is unique) therefore gives $\dim {\mathfrak{t}}_0=\dim {\mathfrak{t}}_0^{\mathrm{\prime }}$ and $\dim {\mathfrak{a}}_0=\dim {\mathfrak{a}}_0^{\mathrm{\prime }}$. Thus the compact and noncompact dimensions are invariants of the conjugacy class of ${\mathfrak{h}}_0$. This is the sense in which the compact and noncompact dimensions are used in the classification of the $\theta$-stable Cartan subalgebras.