Uniqueness and change of positive system in iwasawa decomposition

Action on restricted-root spaces: let $k\in K$, put ${\mathfrak{a}}^{\mathrm{\prime }}=\mathrm{Ad}(k)\mathfrak{a}$ and $H^{\mathrm{\prime }}=\mathrm{Ad}(k)H$ for $H\in \mathfrak{a}$; for $X\in {\mathfrak{g}}_0^{\lambda }$ one has $[H^{\mathrm{\prime }},\mathrm{Ad}(k)X]=\mathrm{Ad}(k)[H,X]=\lambda (H)\mathrm{Ad}(k)X=(\mathrm{Ad}(k)\lambda )(H^{\mathrm{\prime }})\mathrm{Ad}(k)X$, so $\mathrm{Ad}(k){\mathfrak{g}}_0^{\lambda }\subseteq ({\mathfrak{g}}_0^{\mathrm{\prime }}{)}^{\mathrm{Ad}(k)\lambda }$, where $({\mathfrak{g}}_0^{\mathrm{\prime }}{)}^{\nu }=\{Y\in {\mathfrak{g}}_0:[H^{\mathrm{\prime }},Y]=\nu (H^{\mathrm{\prime }})Y\text{ for every }{H}^{\mathrm{\prime }}\in {\mathfrak{a}}^{\mathrm{\prime }}\}$; applying the same inclusion to $k^{-1}$ and $\mathrm{Ad}(k)\lambda$ gives equality, and consequently $\lambda \in \mathrm{\Sigma }({\mathfrak{g}}_0,\mathfrak{a})$ if and only if $\mathrm{Ad}(k)\lambda \in \mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})$, that is, $\mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})=\mathrm{Ad}(k)\mathrm{\Sigma }({\mathfrak{g}}_0,\mathfrak{a})$.

Chambers and positive systems: for a positive system ${\mathrm{\Sigma }}^{+}$ of $\mathrm{\Sigma }$, cut out by a regular $H_0\in \mathfrak{a}$, put $C({\mathrm{\Sigma }}^{+})=\{H\in \mathfrak{a}:\lambda (H)>0\text{ for every }\lambda \in {\mathrm{\Sigma }}^{+}\}$; then $C({\mathrm{\Sigma }}^{+})$ is a nonempty open convex cone contained in the complement of ${\bigcup }_{\lambda \in \mathrm{\Sigma }}{\lambda }^{\perp }$, and it is a full connected component of that complement, because on the finite set $\mathrm{\Sigma }$ the signs $\mathrm{sign}\lambda (H)$ are constant on $C({\mathrm{\Sigma }}^{+})$ and determine the component of $H$; hence $C({\mathrm{\Sigma }}^{+})$ is a chamber, it does not depend on the choice of the regular element $H_0$ cutting out ${\mathrm{\Sigma }}^{+}$, and the assignment ${\mathrm{\Sigma }}^{+}\mapsto C({\mathrm{\Sigma }}^{+})$ is a bijection from the positive systems of $\mathrm{\Sigma }$ onto the chambers, with inverse $C\mapsto \{\lambda \in \mathrm{\Sigma }:\lambda >0\text{ on }C\}$.

Uniqueness of the factorization: if $k_1{a}_1{n}_1=k_2{a}_2{n}_2$ with $k_i\in K$, $a_i\in A$, $n_i\in N$, then injectivity of the multiplication map $K\times A\times N\to G$ forces $(k_1,a_1,n_1)=(k_2,a_2,n_2)$; this proves (a).

Dependence on $\mathfrak{a}$: the exponential map is injective on ${\mathfrak{p}}_0$ by [L1]; distinct maximal abelian subspaces cannot contain one another, so if $\mathfrak{a}\mathrm{\ne }{\mathfrak{a}}^{\mathrm{\prime }}$ we may choose $X\in \mathfrak{a}$ with $X\mathrm{\notin }{\mathfrak{a}}^{\mathrm{\prime }}$; then $\exp X\in A=\exp (\mathfrak{a})$ while $\exp X\mathrm{\notin }\exp ({\mathfrak{a}}^{\mathrm{\prime }})$, because $\exp X=\exp Y$ with $Y\in {\mathfrak{a}}^{\mathrm{\prime }}$ would force $X=Y\in {\mathfrak{a}}^{\mathrm{\prime }}$; hence $A\mathrm{\ne }\exp ({\mathfrak{a}}^{\mathrm{\prime }})$ and $A$ determines $\mathfrak{a}$ and is determined by it.

Dependence on the positive system: by [L6] one has $\theta \mathfrak{n}({\mathrm{\Sigma }}^{+})={⨁}_{\lambda \in {\mathrm{\Sigma }}^{+}}{\mathfrak{g}}_0^{-\lambda }=\mathfrak{n}(-{\mathrm{\Sigma }}^{+})$, and $\mathrm{\Theta }(\exp X)=\exp (\theta X)$ for all $X\in {\mathfrak{g}}_0$, so $\mathrm{\Theta }(N({\mathrm{\Sigma }}^{+}))=N(-{\mathrm{\Sigma }}^{+})$; if $N({\mathrm{\Sigma }}^{+})=N(-{\mathrm{\Sigma }}^{+})$, equality of these Lie subgroups forces equality of their Lie algebras, $\mathfrak{n}({\mathrm{\Sigma }}^{+})=\mathfrak{n}(-{\mathrm{\Sigma }}^{+})$, by [L1]. But these two sums of restricted-root spaces have zero intersection because ${⨁}_{\lambda \in \mathrm{\Sigma }}{\mathfrak{g}}_0^{\lambda }$ is direct and ${\mathrm{\Sigma }}^{+}\cap (-{\mathrm{\Sigma }}^{+})=\mathrm{\varnothing }$; hence both would be zero. Thus $N({\mathrm{\Sigma }}^{+})\mathrm{\ne }N(-{\mathrm{\Sigma }}^{+})$ whenever $\mathfrak{n}({\mathrm{\Sigma }}^{+})\mathrm{\ne }0$. Finally $\mathfrak{n}({\mathrm{\Sigma }}^{+})\mathrm{\ne }0$ when $\mathfrak{a}\mathrm{\ne }0$, because then $\mathrm{\Sigma }\mathrm{\ne }\mathrm{\varnothing }$ since $\mathrm{\Sigma }$ spans ${\mathfrak{a}}^{\ast }$ by [L3], and $\mathrm{\Sigma }={\mathrm{\Sigma }}^{+}\bigsqcup (-{\mathrm{\Sigma }}^{+})$ forces ${\mathrm{\Sigma }}^{+}\mathrm{\ne }\mathrm{\varnothing }$; this proves (b).

Put ${\mathrm{\Sigma }}_s=\{\alpha \in \mathrm{\Sigma }:\alpha \mathrm{/}2\mathrm{\notin }\mathrm{\Sigma }\}$. If $\alpha ,c\alpha \in \mathrm{\Sigma }$ with $c>0$, integrality in both orders gives $2c,2\mathrm{/}c\in \mathbb{Z}$, whose product is four; hence $c\in \{1\mathrm{/}2,1,2\}$. On each root line the shortest positive root is therefore indivisible and the only possible longer root is its double. Thus ${\mathrm{\Sigma }}_s$ has precisely two opposite roots on each root line and spans ${\mathfrak{a}}^{\ast }$. Reflections permute $\mathrm{\Sigma }$ and preserve the condition that half a root is absent, so preserve ${\mathrm{\Sigma }}_s$. Integrality is inherited, proving that ${\mathrm{\Sigma }}_s$ is a reduced crystallographic root system. Since $s_{2\alpha }=s_{\alpha }$, it has the same Weyl group and hyperplanes as $\mathrm{\Sigma }$. Identify ${\mathfrak{a}}^{\ast }$ with $\mathfrak{a}$ by the positive Killing form on $\mathfrak{a}$ to apply the chamber theorem [L5].

We prove conjugacy of maximal abelian subspaces for the full stated group, allowing compact factors. Choose regular $H\in \mathfrak{a}$ and $H^{\mathrm{\prime }}\in {\mathfrak{a}}^{\mathrm{\prime }}$; such points exist outside finitely many proper hyperplanes (and zero is regular in dimension zero). Since $K$ is compact, $k\mapsto B(\mathrm{Ad}(k)H,H^{\mathrm{\prime }})$ has a maximum at $k_0$. Put $X=\mathrm{Ad}(k_0)H$. The adjoint action of $K$ preserves ${\mathfrak{p}}_0$ because it commutes with $\theta$. Differentiating along $\exp (tY)k_0$, for $Y\in {\mathfrak{k}}_0$, and using invariance of the Killing form gives $0=B([Y,X],H^{\mathrm{\prime }})=B(Y,[X,H^{\mathrm{\prime }}])$. Since $[X,H^{\mathrm{\prime }}]\in {\mathfrak{k}}_0$, negative definiteness implies $[X,H^{\mathrm{\prime }}]=0$, hence $X\in {\mathfrak{a}}^{\mathrm{\prime }}$ by [L6]. Thus ${\mathfrak{a}}^{\mathrm{\prime }}\subseteq Z_{{\mathfrak{p}}_0}(X)=\mathrm{Ad}(k_0)\mathfrak{a}$. Maximality of the abelian subspace ${\mathfrak{a}}^{\mathrm{\prime }}$ gives equality.

The restricted Weyl group acts simply transitively on positive systems: an element $k\in N_K(\mathfrak{a})$ preserves $\mathfrak{a}$ and acts on $\mathfrak{a}$ by $w=\mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}$ and on its dual by $w\lambda =\lambda \circ w^{-1}$; by step 1.1 with ${\mathfrak{a}}^{\mathrm{\prime }}=\mathfrak{a}$ it permutes $\mathrm{\Sigma }$, hence permutes the hyperplanes ${\lambda }^{\perp }$ and the chambers; the hyperplanes are those of the reduced crystallographic root system ${\mathrm{\Sigma }}_s$ by step 1.6, whose Weyl group is $W(\mathrm{\Sigma })$, so by [L4] and [L5] the induced action of $W(\mathrm{\Sigma })=W({\mathfrak{g}}_0,\mathfrak{a})$ on the chambers of $\mathrm{\Sigma }$ is simply transitive, and by step 1.2 this is exactly the simply transitive action $w\cdot {\mathrm{\Sigma }}^{+}=w({\mathrm{\Sigma }}^{+})$ on the positive systems.

Change of $\mathfrak{a}$: let $\mathfrak{a},{\mathfrak{a}}^{\mathrm{\prime }}\subseteq {\mathfrak{p}}_0$ be maximal abelian and choose $k\in K$ with $\mathrm{Ad}(k)\mathfrak{a}={\mathfrak{a}}^{\mathrm{\prime }}$ by step 1.7; if the positive system ${\mathrm{\Sigma }}^{+}$ is cut out by the regular element $H_0\in \mathfrak{a}$, then $\mathrm{Ad}(k){\mathrm{\Sigma }}^{+}=\{\nu \in \mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }}):\nu (\mathrm{Ad}(k)H_0)>0\}$, and $\mathrm{Ad}(k)H_0$ is regular for $\mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})$ because $\nu (\mathrm{Ad}(k)H_0)=(\mathrm{Ad}(k{)}^{-1}\nu )(H_0)\mathrm{\ne }0$ for every root $\nu \in \mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})$; hence $\mathrm{Ad}(k){\mathrm{\Sigma }}^{+}$ is a positive system of $\mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})=\mathrm{Ad}(k)\mathrm{\Sigma }({\mathfrak{g}}_0,\mathfrak{a})$ by step 1.1. Moreover $\mathrm{Ad}(k)\mathfrak{n}({\mathrm{\Sigma }}^{+})={⨁}_{\lambda \in {\mathrm{\Sigma }}^{+}}\mathrm{Ad}(k){\mathfrak{g}}_0^{\lambda }={⨁}_{\lambda \in {\mathrm{\Sigma }}^{+}}({\mathfrak{g}}_0^{\mathrm{\prime }}{)}^{\mathrm{Ad}(k)\lambda }=\mathfrak{n}(\mathrm{Ad}(k){\mathrm{\Sigma }}^{+})$ computed with respect to ${\mathfrak{a}}^{\mathrm{\prime }}$, and $k\exp (X)k^{-1}=\exp (\mathrm{Ad}(k)X)$ for all $X\in {\mathfrak{g}}_0$, so conjugation by $k$ carries $A=\exp (\mathfrak{a})$ onto $\exp ({\mathfrak{a}}^{\mathrm{\prime }})$ and $N=\exp (\mathfrak{n}({\mathrm{\Sigma }}^{+}))$ onto $\exp (\mathfrak{n}(\mathrm{Ad}(k){\mathrm{\Sigma }}^{+}))$; this proves (c).

Change of positive system: let ${\mathrm{\Sigma }}^{+},{\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$ be positive systems of $\mathrm{\Sigma }$; by steps 1.2 and 2.1 there is a unique $w\in W(\mathrm{\Sigma })$ with $w({\mathrm{\Sigma }}^{+})={\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$, and by [L4] there is $k\in N_K(\mathfrak{a})$ representing $w$; then step 1.1 gives $\mathrm{Ad}(k){\mathfrak{g}}_0^{\lambda }={\mathfrak{g}}_0^{w\lambda }$ for every $\lambda \in \mathrm{\Sigma }$, hence $\mathrm{Ad}(k)\mathfrak{n}({\mathrm{\Sigma }}^{+})={⨁}_{\lambda \in {\mathrm{\Sigma }}^{+}}{\mathfrak{g}}_0^{w\lambda }=\mathfrak{n}(w{\mathrm{\Sigma }}^{+})=\mathfrak{n}({\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}})$. Conversely, if $k^{\mathrm{\prime }}\in N_K(\mathfrak{a})$ satisfies $\mathrm{Ad}(k^{\mathrm{\prime }})\mathfrak{n}({\mathrm{\Sigma }}^{+})=\mathfrak{n}({\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}})$, then ${⨁}_{\lambda \in {\mathrm{\Sigma }}^{+}}{\mathfrak{g}}_0^{w^{\mathrm{\prime }}\lambda }={⨁}_{\nu \in {\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}}{\mathfrak{g}}_0^{\nu }$ for the class $w^{\mathrm{\prime }}$ of $k^{\mathrm{\prime }}$, and comparing the two direct sums of restricted-root spaces inside the direct sum over $\mathrm{\Sigma }$ gives $\{w^{\mathrm{\prime }}\lambda :\lambda \in {\mathrm{\Sigma }}^{+}\}={\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$, so $w^{\mathrm{\prime }}({\mathrm{\Sigma }}^{+})={\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$ and hence $w^{\mathrm{\prime }}=w$ by uniqueness; therefore $k^{\mathrm{\prime }}\in k{Z}_K(\mathfrak{a})$, by [L4]. With $w\mapsto w({\mathrm{\Sigma }}^{+})$ for the fixed ${\mathrm{\Sigma }}^{+}$ this is a bijection by step 2.1, so the restricted Weyl group permutes the positive systems simply transitively; this proves (d).

Simultaneous change: given $(\mathfrak{a},{\mathrm{\Sigma }}^{+})$ and $({\mathfrak{a}}^{\mathrm{\prime }},{\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}})$, choose $k_1\in K$ with $\mathrm{Ad}(k_1)\mathfrak{a}={\mathfrak{a}}^{\mathrm{\prime }}$ by step 1.7; by step 2.2 the set $\mathrm{Ad}(k_1){\mathrm{\Sigma }}^{+}$ is a positive system of $\mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}^{\mathrm{\prime }})$, and $\mathrm{Ad}(k_1)\mathfrak{n}({\mathrm{\Sigma }}^{+})=\mathfrak{n}(\mathrm{Ad}(k_1){\mathrm{\Sigma }}^{+})$ with respect to ${\mathfrak{a}}^{\mathrm{\prime }}$; by step 3.1 applied to ${\mathfrak{a}}^{\mathrm{\prime }}$ there is $k_2\in N_K({\mathfrak{a}}^{\mathrm{\prime }})$ with $\mathrm{Ad}(k_2)\mathfrak{n}(\mathrm{Ad}(k_1){\mathrm{\Sigma }}^{+})=\mathfrak{n}({\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}})$, equivalently $\mathrm{Ad}(k_2)(\mathrm{Ad}(k_1){\mathrm{\Sigma }}^{+})={\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$; then $k=k_2{k}_1\in K$ satisfies $\mathrm{Ad}(k)\mathfrak{a}={\mathfrak{a}}^{\mathrm{\prime }}$, $\mathrm{Ad}(k){\mathrm{\Sigma }}^{+}={\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}}$ and $\mathrm{Ad}(k)\mathfrak{n}({\mathrm{\Sigma }}^{+})=\mathfrak{n}({\mathrm{\Sigma }}^{{+}^{\mathrm{\prime }}})$, and conjugation by $k$ carries $A$ and $N$ accordingly by step 2.2; this proves (e).

Assertions (a)–(e) are now established by steps 1.3, 1.4 and 1.5, 2.2, 3.1 and 4.1, so all Iwasawa data attached to a maximal abelian $\mathfrak{a}\subseteq {\mathfrak{p}}_0$ and a positive system are unique for fixed data and conjugate by $K$ in general, with the positive systems permuted simply transitively by the restricted Weyl group. The Axiom of Choice was declared in [A1] and is inherited through [L1] and [L6]; only finitely many individual selections occur. If $\mathfrak{a}=0$, the root system is empty, $A=N=\{e\}$ and $W=1$; the single chamber is $\{0\}$ and its positive system is empty, so every assertion includes this case.