Restricted weyl group is the reflection group of the restricted root system

If $\mathfrak{a}=0$ then $\mathrm{\Sigma }=\mathrm{\varnothing }$, because every restricted root is a nonzero element of ${\mathfrak{a}}^{\ast }$ by Restricted root and restricted root space and ${\mathfrak{a}}^{\ast }=0$; moreover $N_K(\mathfrak{a})=\{k\in K:\mathrm{Ad}(k)\mathfrak{a}=\mathfrak{a}\}=K=Z_K(\mathfrak{a})$, so $W({\mathfrak{g}}_0,\mathfrak{a})$ and $W(\mathrm{\Sigma })=⟨\mathrm{\varnothing }⟩$ are both the trivial group and the theorem holds; assume henceforth that $\mathfrak{a}\mathrm{\ne }0$. Setting and notation: $(\cdot ,\cdot )$ is a positive definite inner product on $\mathfrak{a}$ by [L2], the dual vectors $H_{\lambda }$ and the inner product $⟨\cdot ,\cdot ⟩$ on ${\mathfrak{a}}^{\ast }$ are as in the statement, and the reflections $s_{\lambda }$ are orthogonal, fix ${\lambda }^{\perp }$ pointwise and negate $\lambda$; $\mathrm{\Sigma }$ is finite, nonempty and spans ${\mathfrak{a}}^{\ast }$ and satisfies $s_{\lambda }(\mathrm{\Sigma })=\mathrm{\Sigma }$ and the integrality of the Cartan integers by [L1]; put ${\mathrm{\Sigma }}_s=\{\alpha \in \mathrm{\Sigma }:\alpha \mathrm{/}2\mathrm{\notin }\mathrm{\Sigma }\}$ and $\mathrm{\Psi }=\{\alpha \in {\mathrm{\Sigma }}_s:2\alpha \in \mathrm{\Sigma }\}$; then the multiple-set argument with the integrality of [L1] gives $\{c>0:c\lambda \in \mathrm{\Sigma }\}=\{a\}$ or $\{a,2a\}$ with $a\in \{1,\frac{1}{2}\}$ for every $\lambda \in \mathrm{\Sigma }$, hence $\mathrm{\Sigma }={\mathrm{\Sigma }}_s\bigsqcup 2\mathrm{\Psi }$, and for $\beta \in {\mathrm{\Sigma }}_s$ the positive multiples of $\beta$ in $\mathrm{\Sigma }$ are $\beta$ and possibly $2\beta$; consequently ${\mathrm{\Sigma }}_s$ is a reduced crystallographic root system in ${\mathfrak{a}}^{\ast }$ with $W({\mathrm{\Sigma }}_s)=W(\mathrm{\Sigma })$, and the hyperplanes ${\lambda }^{\perp }$ for $\lambda \in \mathrm{\Sigma }$ are exactly the hyperplanes ${\alpha }^{\perp }$ for $\alpha \in {\mathrm{\Sigma }}_s$; also choose a chamber $C$ of the arrangement (nonempty by [L3]), let ${\mathrm{\Sigma }}^{+}=\{\lambda \in \mathrm{\Sigma }:\lambda >0$ on $C\}$ and ${\mathrm{\Sigma }}_s^{+}={\mathrm{\Sigma }}_s\cap {\mathrm{\Sigma }}^{+}$, put $\delta =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Sigma }}_s^{+}}\alpha$ and let $H_{\delta }\in \mathfrak{a}$ be its dual vector, so that $\lambda (H_{\delta })=⟨\lambda ,\delta ⟩$ for all $\lambda \in {\mathfrak{a}}^{\ast }$.

The compact form: put $\mathfrak{u}={\mathfrak{k}}_0\oplus i{\mathfrak{p}}_0\subseteq {\mathfrak{g}}_0^{\mathbb{C}}$; it is a real Lie subalgebra with ${\mathfrak{u}}^{\mathbb{C}}={\mathfrak{g}}_0^{\mathbb{C}}$ and ${\dim }_{\mathbb{R}}\mathfrak{u}={\dim }_{\mathbb{C}}{\mathfrak{g}}_0^{\mathbb{C}}$, hence a real form; its Killing form is the restriction of the Killing form of ${\mathfrak{g}}_0^{\mathbb{C}}$ and is negative definite, since it equals the Killing form of ${\mathfrak{g}}_0$ on ${\mathfrak{k}}_0$, where the latter is negative definite, and equals its negative on $i{\mathfrak{p}}_0$, where the Killing form of ${\mathfrak{g}}_0$ is positive definite by [L2], with the two summands orthogonal; hence $\mathfrak{u}$ is a compact real form, and $\mathfrak{u}$ is semisimple because ${\mathfrak{u}}^{\mathbb{C}}={\mathfrak{g}}_0^{\mathbb{C}}$ is semisimple by [L7].

$W(\mathrm{\Sigma })\subseteq W({\mathfrak{g}}_0,\mathfrak{a})$: by [L1] each reflection $s_{\lambda }$ is realised by some $k_{\lambda }\in N_K(\mathfrak{a})$, whose class in $W({\mathfrak{g}}_0,\mathfrak{a})$ acts on ${\mathfrak{a}}^{\ast }$ by $s_{\lambda }$; the classes of the $k_{\lambda }$ therefore generate a subgroup of $W({\mathfrak{g}}_0,\mathfrak{a})$ that is identified with $W(\mathrm{\Sigma })=⟨s_{\lambda }:\lambda \in \mathrm{\Sigma }⟩$.

$W({\mathfrak{g}}_0,\mathfrak{a})$ permutes the restricted roots and the chambers: if $k\in N_K(\mathfrak{a})$, $\mu \in \mathrm{\Sigma }$ and $0\mathrm{\ne }X\in {\mathfrak{g}}_0^{\mu }$, then for $H\in \mathfrak{a}$ one has $[H,\mathrm{Ad}(k)X]=\mathrm{Ad}(k)[\mathrm{Ad}(k{)}^{-1}H,X]=({\lambda }_k(\mu ))(H)\mathrm{Ad}(k)X$ with ${\lambda }_k(\mu )=\mu \circ \mathrm{Ad}(k{)}^{-1}$, so $\mathrm{Ad}(k)$ carries ${\mathfrak{g}}_0^{\mu }$ onto ${\mathfrak{g}}_0^{{\lambda }_k(\mu )}$ and ${\lambda }_k(\mu )\in \mathrm{\Sigma }$; hence the dual action of $W({\mathfrak{g}}_0,\mathfrak{a})$ permutes $\mathrm{\Sigma }$, and therefore permutes the hyperplanes ${\lambda }^{\perp }$ and the chambers of their complement.

Regularity of $H_{\delta }$: by [L4] applied to the reduced system ${\mathrm{\Sigma }}_s$ with its positive system ${\mathrm{\Sigma }}_s^{+}$ one has $⟨\delta ,\alpha ⟩={\sum }_i{n}_i⟨{\alpha }_i,{\omega }_i⟩={\sum }_i{n}_i({\alpha }_i,{\alpha }_i)\mathrm{/}2$ for $\alpha ={\sum }_i{n}_i{\alpha }_i\in {\mathrm{\Sigma }}_s^{+}$ over the simple roots, so $⟨\delta ,\alpha ⟩>0$ for $\alpha \in {\mathrm{\Sigma }}_s^{+}$ and $⟨\delta ,\alpha ⟩<0$ for $\alpha \in {\mathrm{\Sigma }}_s^{-}$; hence $⟨\delta ,\lambda ⟩\mathrm{\ne }0$ for every $\lambda \in \mathrm{\Sigma }$, because every $\lambda$ is $\alpha$ or $2\alpha$ with $\alpha \in {\mathrm{\Sigma }}_s\setminus \{0\}$ by step 1.1, and therefore $\lambda (H_{\delta })\mathrm{\ne }0$ for every $\lambda \in \mathrm{\Sigma }$; also $\mathrm{\Sigma }\subseteq \mathrm{span}{\mathrm{\Sigma }}_s$ by step 1.1.

The group $\tilde{U}=\mathrm{Aut}(\mathfrak{u}{)}^0$ is a compact connected Lie group with Lie algebra $\mathrm{Der}(\mathfrak{u})=\mathrm{ad}(\mathfrak{u})$: the automorphism group is closed in $\mathrm{GL}(\mathfrak{u})$ because the conditions $\phi [x,y]=[\phi x,\phi y]$ are polynomial, it preserves the Killing form of $\mathfrak{u}$, so it lies in the orthogonal group of that negative definite form and is compact, and its identity component is therefore a compact connected Lie group; by [L6] its Lie algebra is $\mathrm{Der}(\mathfrak{u})=\mathrm{ad}(\mathfrak{u})$, with $\mathrm{ad}$ injective by [L7].

Transitivity on chambers: by step 1.1 the chambers of the arrangement of $\mathrm{\Sigma }$ are the open Weyl chambers of the reduced crystallographic root system ${\mathrm{\Sigma }}_s$, and $W(\mathrm{\Sigma })=W({\mathrm{\Sigma }}_s)$, so [L3] shows that $W(\mathrm{\Sigma })$ acts simply transitively, in particular transitively, on these chambers.

The Cartan decomposition in [L2] is a diffeomorphism $K\times {\mathfrak{p}}_0\to G$. Since $G$ and ${\mathfrak{p}}_0$ are connected, $K$ is connected; a connected Lie group is generated by exponentials of its Lie algebra. Thus for every $k\in K$ the restriction to $\mathfrak{u}$ of the complex-linear extension of $\mathrm{Ad}(k)$ lies in $\tilde{U}$: $\mathrm{Ad}(k)$ preserves ${\mathfrak{k}}_0$ and ${\mathfrak{p}}_0$ by [L2], so it preserves $\mathfrak{u}={\mathfrak{k}}_0\oplus i{\mathfrak{p}}_0$, and writing $k=\exp X_1\cdots \exp X_m$ with $X_j\in {\mathfrak{k}}_0$ exhibits the restriction as $\exp ({\mathrm{ad}}_{\mathfrak{u}}{X}_1)\cdots \exp ({\mathrm{ad}}_{\mathfrak{u}}{X}_m)$, a product of exponentials of inner derivations, hence an element of the identity component $\tilde{U}$.

Reduction to a positive-system stabilizer: let $v\in W({\mathfrak{g}}_0,\mathfrak{a})$ be represented by $k\in N_K(\mathfrak{a})$ and let $C$ be the chamber of step 1.1 with positive system ${\mathrm{\Sigma }}^{+}$; by steps 2.3 and 1.4 the set $v(C)$ is a chamber, so there is $u\in W(\mathrm{\Sigma })\subseteq W({\mathfrak{g}}_0,\mathfrak{a})$ with $u(v(C))=C$; representing $u$ by some $k^{\mathrm{\prime }}\in N_K(\mathfrak{a})$ by [L1] and putting $k^{\mathrm{\prime }\mathrm{\prime }}=k^{\mathrm{\prime }}k$, one has $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})({\mathrm{\Sigma }}^{+})={\mathrm{\Sigma }}^{+}$; since $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})$ permutes ${\mathrm{\Sigma }}_s^{+}={\mathrm{\Sigma }}_s\cap {\mathrm{\Sigma }}^{+}$, it fixes $\delta =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Sigma }}_s^{+}}\alpha$, and since by [L2] it preserves the inner product on $\mathfrak{a}$ it also fixes the dual vector $H_{\delta }$.

The element $\rho \in \tilde{U}$ given by the restriction to $\mathfrak{u}$ of the complex-linear extension of $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})$ lies in $\tilde{U}$ by step 3.1 and centralizes the torus $S=\stackrel{‾}{\{\exp (t\cdot i{\mathrm{ad}}_{\mathfrak{u}}{H}_{\delta }):t\in \mathbb{R}\}}\subseteq \tilde{U}$: indeed ${\mathrm{ad}}_{\mathfrak{u}}(i{H}_{\delta })\in \mathrm{ad}(\mathfrak{u})=\mathrm{Lie}\tilde{U}$ by step 2.2, so the closure of the one-parameter subgroup is a compact connected abelian, hence torus, subgroup of $\tilde{U}$, and $\rho {\mathrm{ad}}_{\mathfrak{u}}(i{H}_{\delta }){\rho }^{-1}={\mathrm{ad}}_{\mathfrak{u}}(\rho (i{H}_{\delta }))={\mathrm{ad}}_{\mathfrak{u}}(i{H}_{\delta })$ because $\rho$ fixes $H_{\delta }$ by step 3.2 and the restriction to $\mathfrak{u}$ of the complex-linear extension of $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})$ sends $i{H}_{\delta }$ to $i{H}_{\delta }$; exponentiating gives $\rho \in Z_{\tilde{U}}(S)$.

The Lie algebra of $Z_{\tilde{U}}(S)$ is ${\mathrm{ad}}_{\mathfrak{u}}({\mathfrak{z}}_{\mathfrak{u}}({\mathfrak{s}}^{\mathrm{\prime }}))$ where ${\mathfrak{s}}^{\mathrm{\prime }}={\mathrm{ad}}_{\mathfrak{u}}^{-1}(\mathrm{Lie}S)$: by [L5] the closed subgroup $Z_{\tilde{U}}(S)$ is connected (the centralizer of the torus $S$ in the compact connected group $\tilde{U}$), and its Lie algebra is $\{Y\in \mathrm{ad}(\mathfrak{u}):[Y,\mathrm{Lie}S]=0\}$, which is ${\mathrm{ad}}_{\mathfrak{u}}(\{X\in \mathfrak{u}:[X,{\mathfrak{s}}^{\mathrm{\prime }}]=0\})$ because $[Y,{\mathrm{ad}}_{\mathfrak{u}}X]={\mathrm{ad}}_{\mathfrak{u}}[Y,X]$ and ${\mathrm{ad}}_{\mathfrak{u}}$ is injective; here $i{H}_{\delta }\in {\mathfrak{s}}^{\mathrm{\prime }}$ by step 4.1; and ${\mathfrak{z}}_{\mathfrak{u}}({\mathfrak{s}}^{\mathrm{\prime }})\subseteq {\mathfrak{z}}_{\mathfrak{u}}(i{H}_{\delta })=\mathfrak{u}\cap Z_{{\mathfrak{g}}_0^{\mathbb{C}}}(H_{\delta })=\mathfrak{u}\cap (\mathfrak{a}\oplus \mathfrak{m}{)}^{\mathbb{C}}=\mathfrak{m}\oplus i\mathfrak{a}$, using $Z_{{\mathfrak{g}}_0}(H_{\delta })={\mathfrak{g}}_0^0$ of [L2] (legitimate by the regularity proved in step 2.1) and the decomposition ${\mathfrak{g}}_0^0=\mathfrak{a}\oplus \mathfrak{m}$.

Consequently $W({\mathfrak{g}}_0,\mathfrak{a})\subseteq W(\mathrm{\Sigma })$: every element of $\mathfrak{m}\oplus i\mathfrak{a}$ commutes with $i\mathfrak{a}$, because $\mathfrak{m}=Z_{{\mathfrak{k}}_0}(\mathfrak{a})$ and $\mathfrak{a}$ is abelian, so by step 5.1 every element of the connected group $Z_{\tilde{U}}(S)$, which is generated by the exponentials of its Lie algebra, fixes $i\mathfrak{a}$ pointwise; hence $\rho$ fixes $i\mathfrak{a}$ pointwise by step 4.1, and therefore $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})$ fixes $\mathfrak{a}$ pointwise, that is $k^{\mathrm{\prime }\mathrm{\prime }}\in Z_K(\mathfrak{a})$; since $\mathrm{Ad}(k^{\mathrm{\prime }\mathrm{\prime }})=\mathrm{Ad}(k^{\mathrm{\prime }})\mathrm{Ad}(k)$, the class $v$ of $k$ satisfies $v=\mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}=\mathrm{Ad}(k^{\mathrm{\prime }}{)}^{-1}{\mathrm{\mid }}_{\mathfrak{a}}\in W(\mathrm{\Sigma })$.

Combining steps 1.3 and 6.1 gives $W({\mathfrak{g}}_0,\mathfrak{a})=W(\mathrm{\Sigma })$; since $W(\mathrm{\Sigma })=W({\mathrm{\Sigma }}_s)$ is the Weyl group of the reduced crystallographic root system ${\mathrm{\Sigma }}_s$ by step 1.1, it is finite by [L3], and by construction it is generated by the orthogonal reflections $s_{\lambda }$ in the restricted-root hyperplanes; the Axiom of Choice was used through the compact-group connectedness theorem of [L5] and through the restricted-root theory of [L1].