Restricted root space decomposition

For $H\in {\mathfrak{p}}_0$ the endomorphism $\mathrm{ad}H$ of ${\mathfrak{g}}_0$ is self-adjoint for $B_{\theta }$: for all $X,Y\in {\mathfrak{g}}_0$, using $\theta H=-H$ and [L1], $B_{\theta }([H,X],Y)=-B([H,X],\theta Y)=-B(X,[\theta Y,H])=B(X,[H,\theta Y])=B_{\theta }(X,[H,Y])$; moreover $\mathrm{ad}[H,H^{\mathrm{\prime }}]=[\mathrm{ad}H,\mathrm{ad}{H}^{\mathrm{\prime }}]$, so for $H,H^{\mathrm{\prime }}\in \mathfrak{a}$ the endomorphisms $\mathrm{ad}H$ and $\mathrm{ad}{H}^{\mathrm{\prime }}$ commute because $[H,H^{\mathrm{\prime }}]=0$, and hence $\{\mathrm{ad}H:H\in \mathfrak{a}\}$ is a commuting family of self-adjoint, therefore diagonalisable, endomorphisms of ${\mathfrak{g}}_0$.

If $0\mathrm{\ne }X\in {\mathfrak{g}}_0$ is a common eigenvector of the family $\{\mathrm{ad}H:H\in \mathfrak{a}\}$, define ${\lambda }_X:\mathfrak{a}\to \mathbb{R}$ by $\mathrm{ad}H(X)={\lambda }_X(H)X$ for $H\in \mathfrak{a}$; then ${\lambda }_X$ is linear, because for $H,H^{\mathrm{\prime }}\in \mathfrak{a}$ and $c\in \mathbb{R}$ one has ${\lambda }_X(H+c{H}^{\mathrm{\prime }})X=\mathrm{ad}(H+c{H}^{\mathrm{\prime }})X=({\lambda }_X(H)+c{\lambda }_X(H^{\mathrm{\prime }}))X$ and $X\mathrm{\ne }0$ permits cancellation of $X$.

Bracket relation: for $X\in {\mathfrak{g}}_0^{\lambda }$, $Y\in {\mathfrak{g}}_0^{\mu }$ and $H\in \mathfrak{a}$, the Jacobi identity gives ${\mathrm{ad}}_H([X,Y])=[{\mathrm{ad}}_{H}X,Y]+[X,{\mathrm{ad}}_{H}Y]=(\lambda (H)+\mu (H))[X,Y]=(\lambda +\mu )(H)[X,Y]$, so $[X,Y]\in {\mathfrak{g}}_0^{\lambda +\mu }$; in particular $[{\mathfrak{g}}_0^{\lambda },{\mathfrak{g}}_0^{\mu }]\subseteq {\mathfrak{g}}_0^{\lambda +\mu }$.

$\theta$-stability: for $X\in {\mathfrak{g}}_0^{\lambda }$ and $H\in \mathfrak{a}$ one has $\theta H=-H$ and $\theta$ is an automorphism, so $[H,\theta X]=\theta [\theta H,X]=\theta [-H,X]=-\theta [H,X]=-\lambda (H)\theta X$, that is $\theta X\in {\mathfrak{g}}_0^{-\lambda }$; since $\theta$ is an involution, $\theta {\mathfrak{g}}_0^{\lambda }={\mathfrak{g}}_0^{-\lambda }$, and ${\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0$ exactly when ${\mathfrak{g}}_0^{-\lambda }\mathrm{\ne }0$.

$\mathfrak{a}={\mathfrak{p}}_0\cap {\mathfrak{g}}_0^0$: the inclusion $\mathfrak{a}\subseteq {\mathfrak{p}}_0\cap {\mathfrak{g}}_0^0$ holds because $\mathfrak{a}\subseteq {\mathfrak{p}}_0$ and $\mathfrak{a}$ is abelian; conversely, if $X\in {\mathfrak{p}}_0\cap {\mathfrak{g}}_0^0$, then $[X,\mathfrak{a}]=0$ and $[X,X]=0$, so $\mathfrak{a}+\mathbb{R}X$ is an abelian subspace of ${\mathfrak{p}}_0$ containing $\mathfrak{a}$, and maximality of $\mathfrak{a}$ forces $X\in \mathfrak{a}$.

Choose a finite basis $H_1,\dots ,H_r$ of $\mathfrak{a}$. By [L2] the commuting operators $\mathrm{ad}{H}_1,\dots ,\mathrm{ad}{H}_r$ have a common eigenbasis $\{X_1,\dots ,X_N\}$ of ${\mathfrak{g}}_0$. Every $H\in \mathfrak{a}$ is a linear combination of the $H_j$, so each $X_i$ is an eigenvector of $\mathrm{ad}H$ as well and lies in ${\mathfrak{g}}_0^{{\lambda }_{X_i}}$. Thus the subspaces ${\mathfrak{g}}_0^{\lambda }$ span ${\mathfrak{g}}_0$ and only finitely many functionals $\lambda \in {\mathfrak{a}}^{\ast }$ occur with ${\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0$. This includes $\mathfrak{a}=0$, when the operator family is empty.

Orthogonality: if $\lambda \mathrm{\ne }\mu$ are occurring functionals, there is $H\in \mathfrak{a}$ with $\lambda (H)\mathrm{\ne }\mu (H)$; the spaces ${\mathfrak{g}}_0^{\lambda }$ and ${\mathfrak{g}}_0^{\mu }$ are eigenspaces of the self-adjoint endomorphism $\mathrm{ad}H$ for the distinct real eigenvalues $\lambda (H)$ and $\mu (H)$, hence are orthogonal for $B_{\theta }$ by [L3].

${\mathfrak{g}}_0^0=\mathfrak{a}\oplus \mathfrak{m}$: by step 1.4 with $\lambda =0$ the involution $\theta$ preserves ${\mathfrak{g}}_0^0$, so every $X\in {\mathfrak{g}}_0^0$ decomposes as $X=\frac{1}{2}(X+\theta X)+\frac{1}{2}(X-\theta X)$ with the first summand in ${\mathfrak{k}}_0\cap {\mathfrak{g}}_0^0$ and the second in ${\mathfrak{p}}_0\cap {\mathfrak{g}}_0^0=\mathfrak{a}$ by step 1.5; also ${\mathfrak{k}}_0\cap {\mathfrak{g}}_0^0={\mathfrak{k}}_0\cap Z_{{\mathfrak{g}}_0}(\mathfrak{a})=Z_{{\mathfrak{k}}_0}(\mathfrak{a})=\mathfrak{m}$, so ${\mathfrak{g}}_0^0=\mathfrak{m}\oplus \mathfrak{a}$, the sum is direct because ${\mathfrak{k}}_0\cap {\mathfrak{p}}_0=0$, and it is orthogonal by [L1].

Conversely, if an occurring functional $\lambda$ is given, that is ${\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0$, pick $0\mathrm{\ne }X\in {\mathfrak{g}}_0^{\lambda }$; then $X$ is a common eigenvector with ${\lambda }_X=\lambda$, so the set of functionals $\lambda \in {\mathfrak{a}}^{\ast }$ with ${\mathfrak{g}}_0^{\lambda }\mathrm{\ne }0$ is exactly the finite set of functionals ${\lambda }_{X_i}$ of step 2.1, and $\mathrm{\Sigma }$ is the set of its nonzero members: $\mathrm{\Sigma }$ is finite and each multiplicity $m_{\lambda }={\dim }_{\mathbb{R}}{\mathfrak{g}}_0^{\lambda }$ is a finite positive integer.

The occurring spaces ${\mathfrak{g}}_0^{\lambda }$ are linearly independent and span ${\mathfrak{g}}_0$: if ${\sum }_{\lambda }{X}_{\lambda }=0$ with $X_{\lambda }\in {\mathfrak{g}}_0^{\lambda }$ and some $X_{{\lambda }_0}\mathrm{\ne }0$, expand in the common eigenbasis of step 2.1; a basis vector with functional ${\lambda }_{X_i}$ can appear with nonzero coefficient in $X_{{\lambda }_0}$ only when ${\lambda }_{X_i}={\lambda }_0$, and distinct occurring functionals involve disjoint groups of basis vectors, so ${\sum }_{\lambda }{X}_{\lambda }\mathrm{\ne }0$, a contradiction; hence ${\mathfrak{g}}_0={⨁}_{\lambda \in {\mathfrak{a}}^{\ast }}{\mathfrak{g}}_0^{\lambda }$, the sum extending over the finitely many occurring functionals, and discarding $\lambda =0$ while using that the occurring nonzero functionals are exactly the elements of $\mathrm{\Sigma }$ and that ${\mathfrak{g}}_0^0=Z_{{\mathfrak{g}}_0}(\mathfrak{a})$ by Restricted root and restricted root space gives the direct sum of statement 1.

If $H\in \mathfrak{a}$ satisfies $\lambda (H)\mathrm{\ne }0$ for every $\lambda \in \mathrm{\Sigma }$, then $Z_{{\mathfrak{g}}_0}(H)={⨁}_{\lambda :\lambda (H)=0}{\mathfrak{g}}_0^{\lambda }={\mathfrak{g}}_0^0$, because ${\mathfrak{g}}_0^{\lambda }$ lies in the kernel of $\mathrm{ad}H$ exactly when $\lambda (H)=0$.

Statements 1–5 are now established: the direct-sum decomposition in step 4.1; the finiteness of $\mathrm{\Sigma }$ and of the multiplicities in step 3.1; the orthogonality in step 2.2; the description of ${\mathfrak{g}}_0^0$ in steps 1.5 and 2.3; the bracket relation in step 1.3; the $\theta$-stability in step 1.4; and the regular-element centralizer in step 5.1. The Axiom of Choice was declared in [A1], and no selection was made in the argument.