Restricted weyl group

Assume the Axiom of Choice (The Axiom of Choice). Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$, let $G$ be a connected semisimple Lie group with finite center and Lie algebra ${\mathfrak{g}}_0$, let $\mathrm{\Theta }$ be a global Cartan involution of $G$ with $d{\mathrm{\Theta }}_e=\theta$, and put $K=G^{\mathrm{\Theta }}$ (Global Cartan decomposition for a connected finite center semisimple Lie group); then $K$ is a closed compact subgroup of $G$ with Lie algebra ${\mathfrak{k}}_0$, and we write $\mathrm{Ad}$ for the adjoint representation of $G$. Let $\mathfrak{a}\subseteq {\mathfrak{p}}_0$ be a maximal abelian subspace (Maximal split abelian subspace and real rank). The normalizer and centralizer of $\mathfrak{a}$ in $K$ are $N_K(\mathfrak{a})=\{k\in K:\mathrm{Ad}(k)\mathfrak{a}=\mathfrak{a}\},Z_K(\mathfrak{a})=\{k\in K:\mathrm{Ad}(k)H=H\text{ for every }H\in \mathfrak{a}\}\mathrm{.}$ Both are subgroups of $K$: they contain the identity, and for $k,l\in K$ the identities $\mathrm{Ad}(kl)=\mathrm{Ad}(k)\mathrm{Ad}(l)$ and $\mathrm{Ad}(k^{-1})=\mathrm{Ad}(k{)}^{-1}$ show stability under products and inverses. They are closed in $K$, being the preimages of the closed conditions $\mathrm{Ad}(k)\mathfrak{a}=\mathfrak{a}$ and $\mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}=\mathrm{i}\mathrm{d}$ under the continuous homomorphism $\mathrm{Ad}$; and $Z_K(\mathfrak{a})$ is a normal subgroup of $N_K(\mathfrak{a})$, because it is the kernel of the restriction $N_K(\mathfrak{a})⟶\mathrm{GL}(\mathfrak{a}),k⟼\mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}\mathrm{.}$

The restricted Weyl group of the pair $({\mathfrak{g}}_0,\mathfrak{a})$ is the quotient group $W({\mathfrak{g}}_0,\mathfrak{a})=N_K(\mathfrak{a})\mathrm{/}{Z}_K(\mathfrak{a})\mathrm{.}$ For $k\in N_K(\mathfrak{a})$ the restriction $\mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}$ is a linear automorphism of $\mathfrak{a}$, since $\mathrm{Ad}(k)$ is invertible and preserves $\mathfrak{a}$; the assignment $k\mapsto \mathrm{Ad}(k){\mathrm{\mid }}_{\mathfrak{a}}$ is a homomorphism by the identities above, so it descends to a well-defined injective homomorphism $W({\mathfrak{g}}_0,\mathfrak{a})⟶\mathrm{GL}(\mathfrak{a}),$ whose injectivity is exactly the definition of $Z_K(\mathfrak{a})$ as kernel. Thus $W({\mathfrak{g}}_0,\mathfrak{a})$ is a group of linear transformations of $\mathfrak{a}$, and it acts faithfully on $\mathfrak{a}$; since $\mathrm{Ad}(k)$ is invertible, the transpose action $(\lambda ,k)⟼\lambda \circ \mathrm{Ad}(k{)}^{-1},{\mathfrak{a}}^{\ast }\times W({\mathfrak{g}}_0,\mathfrak{a})⟶{\mathfrak{a}}^{\ast },$ gives a faithful dual action on the dual space ${\mathfrak{a}}^{\ast }$. The notation $W({\mathfrak{g}}_0,\mathfrak{a})$ is the one used in Restricted weyl group is the reflection group of the restricted root system: its elements act on the restricted roots $\mathrm{\Sigma }=\mathrm{\Sigma }({\mathfrak{g}}_0,\mathfrak{a})$ (Restricted root and restricted root space) by permuting them, and the restricted Weyl group is the reflection group of $\mathrm{\Sigma }$, so it is a finite group; the reflection group structure is the subject of that item.