Analytic and root-system Weyl groups agree

Put $\mathfrak{s}=[\mathfrak{g},\mathfrak{g}]$, $\mathfrak{z}=\mathfrak{z}(\mathfrak{g})$ and ${\mathfrak{t}}_s=\mathfrak{t}\cap \mathfrak{s}$. The invariant inner product in [L10] makes the orthogonal complement of every ideal an ideal, so finite-dimensional induction makes the adjoint module completely reducible; [L10] therefore gives $\mathfrak{g}=\mathfrak{z}\oplus \mathfrak{s}$ with $\mathfrak{s}$ semisimple. We next identify the exact Lie-algebra root interface rather than assuming it from [L1]. If $X$ centralizes $\mathfrak{t}$, then [L11] gives $\mathrm{Ad}(\exp (sX))H=H$ for every $H\in \mathfrak{t}$; naturality and an exponential identity neighborhood show that $\exp (sX)$ centralizes $T$, so [L4] puts it in $T$ and differentiation gives $X\in \mathfrak{t}$. Thus ${\mathfrak{c}}_{\mathfrak{g}}(\mathfrak{t})=\mathfrak{t}$, whence $\mathfrak{z}\subseteq \mathfrak{t}$ and $\mathfrak{t}=\mathfrak{z}\oplus {\mathfrak{t}}_s$. The operators ${\mathrm{ad}}_H$ for $H\in {\mathfrak{t}}_s$ are commuting skew-adjoint operators, so their complexifications are simultaneously diagonalizable. If $U+iV\in {\mathfrak{s}}_{\mathbb{C}}$ centralizes $({\mathfrak{t}}_s{)}_{\mathbb{C}}$, then $U,V\in \mathfrak{s}$ centralize ${\mathfrak{t}}_s$ and also $\mathfrak{z}$, hence all of $\mathfrak{t}$; the centralizer equality puts $U,V$ in ${\mathfrak{t}}_s$. Therefore $({\mathfrak{t}}_s{)}_{\mathbb{C}}$ is maximal toral, hence Cartan by [L11]. Finally the adjoint $T$-weight decomposition in [L1], after differentiation, has zero space ${\mathfrak{t}}_{\mathbb{C}}$ and nonzero weights trivial on $\mathfrak{z}$; restricting it to ${\mathfrak{s}}_{\mathbb{C}}$ gives exactly the root-space decomposition relative to this Cartan algebra. Thus every root used below is a Lie-algebra root to which [L2] and [L7] apply. The invariant inner product makes every ${\mathrm{ad}}_X$, $X\in \mathfrak{s}$, skew-adjoint; in an orthonormal real basis, $B(X,X)=\mathrm{tr}({\mathrm{ad}}_X^2)=-{\sum }_{j,k}({\mathrm{ad}}_X{)}_{jk}^2\le 0.$ Equality forces ${\mathrm{ad}}_X=0$, hence $X=0$ because $\mathfrak{s}$ is centerless by [L10]. Thus $B$ is negative definite on $\mathfrak{s}$ and its complex-bilinear extension is positive definite on $i{\mathfrak{t}}_s$. For a root use [L2] to choose $e,f,h$. Conjugation $\sigma$ sends its root space to the opposite one because the root is imaginary on $\mathfrak{t}$, and sends $h\in i{\mathfrak{t}}_s$ to $-h$. By [L7], $\sigma (e)=cf$ for a nonzero scalar. Invariance of $B$ gives $B(h,h)=B([e,f],h)=2B(e,f)>0$. Writing $e=U+iV$ with $U,V\in \mathfrak{s}$, we have $B(e,\sigma e)=B(U,U)+B(V,V)<0$. Therefore $c=B(e,\sigma e)\mathrm{/}B(e,f)$ is real and negative. Replacing $(e,f)$ by $(ae,a^{-1}f)$ with $\mathrm{\mid }a{\mathrm{\mid }}^2=-1\mathrm{/}c$ gives $\sigma (e)=-f$, and involutivity gives $\sigma (f)=-e$. Thus $X=e-f$, $Y=i(e+f)$ and $H=ih$ belong to $\mathfrak{s}$ and satisfy $[H,X]=2Y$, $[H,Y]=-2X$, $[X,Y]=2H$. They are the images of the standard traceless skew-Hermitian basis of $\mathfrak{s}\mathfrak{u}(2)$.

The map $(z,w)\mapsto \left(\begin{array}{cc}z& -\bar{w}\\ w& \bar{z}\end{array}\right)$ identifies the unit sphere $\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=1$ in ${\mathbb{C}}^2$ homeomorphically with $SU(2)$: orthonormality of the columns and determinant one force the displayed second column, and the inverse reads the first column. Thus $SU(2)$ is connected and simply connected by [L8]. A connected Lie group is generated by any exponential neighborhood of the identity: the generated subgroup is open and all its cosets are open, so connectedness forces it to be the whole group.

By [L3] the inclusion $\mathfrak{s}\mathfrak{u}(2)\cong \mathfrak{k}\hookrightarrow \mathfrak{g}$ integrates to a Lie-group homomorphism ${\mathrm{\Phi }}_{\alpha }:SU(2)\to G$. Define ${\alpha }^{\vee }:S^1\to G$ by ${\alpha }^{\vee }(z):={\mathrm{\Phi }}_{\alpha }(\mathrm{diag}(z,z^{-1}))$. Since $\mathrm{diag}(e^{i\theta },e^{-i\theta })=\exp (\theta \cdot \mathrm{diag}(i,-i))$ and $\mathrm{diag}(i,-i)$ corresponds to $H\in \mathfrak{t}$, the image of ${\alpha }^{\vee }$ is $\exp (\mathbb{R}H)\subseteq T$, so ${\alpha }^{\vee }$ is a cocharacter of $T$.

On the root space ${\mathfrak{g}}_{\alpha }$ the element $H$ acts by the scalar $\alpha (H)=i\alpha (h_{\alpha })=2i$, so $\exp (\theta H)$ acts on ${\mathfrak{g}}_{\alpha }$ by $e^{2i\theta }$, and ${\mathrm{\Phi }}_{\alpha }(\mathrm{diag}(z,z^{-1}))$ acts on ${\mathfrak{g}}_{\alpha }$ by $z^2$; therefore $\alpha ({\alpha }^{\vee }(z))=z^2$ for all $z$, that is, $⟨\alpha ,{\alpha }^{\vee }⟩=2$.

The standard matrix $n_0=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ equals $\exp (-\pi X_0\mathrm{/}2)$, where $X_0=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. It conjugates $\mathrm{diag}(i,-i)$ to its negative. If $Z\in \mathfrak{t}$ satisfies $\alpha (Z)=0$, the root relations give $[Z,e]=[Z,f]=0$, so $[Z,X]=0$ and [L9] gives $\mathrm{Ad}({\mathrm{\Phi }}_{\alpha }(n_0))Z=Z$. Hence $n_{\alpha }={\mathrm{\Phi }}_{\alpha }(n_0)$ fixes $\ker \alpha$ pointwise and negates $H$. Since $\alpha (H)=2i$, these subspaces give all of $\mathfrak{t}$, and the action is the coroot reflection. Naturality and generation by exponential neighborhoods (step 1.2) give $n_{\alpha }T{n}_{\alpha }^{-1}=T$. Every element of connected $G$ fixes $\mathfrak{z}(\mathfrak{g})$ under the adjoint action: this holds on exponentials since ${\mathrm{ad}}_Z$ kills the center, and hence on their generated group. Thus the faithful action in [L4] stays faithful on ${\mathfrak{t}}_s$, and the realized reflections give $W(\mathrm{\Phi })\subseteq W(G,T)$.

Conversely let $g\in N_G(T)$ with class $w\in W(G,T)$. Conjugation sends a root vector of weight $\alpha$ to one of weight $\alpha \circ \mathrm{Ad}(g{)}^{-1}$, so it permutes the roots. It preserves $B$ because it conjugates adjoint matrices and trace is invariant under conjugation. Therefore it takes positive systems to positive systems. By [L5] there is $w^{\mathrm{\prime }}\in W(\mathrm{\Phi })$ with $w^{\mathrm{\prime }}w({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{+}$; by step 3.2 the element $w^{\mathrm{\prime }}$ is realised by some $n^{\mathrm{\prime }}\in N_G(T)$, so $a:=n^{\mathrm{\prime }}g$ satisfies $\mathrm{Ad}(a)({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{+}$.

Because $a$ permutes the positive roots, it fixes their half-sum $\rho$. Let $H_{\rho }\in i{\mathfrak{t}}_s$ be the $B$-dual of $\rho$; invariance of $B$ gives $\mathrm{Ad}(a)H_{\rho }=H_{\rho }$. Set $V_{\rho }=i{H}_{\rho }\in {\mathfrak{t}}_s$. For each simple root, [L6] gives $(\rho ,{\alpha }_i)=\mathrm{\mid }{\alpha }_i{\mathrm{\mid }}^2\mathrm{/}2>0$. Every positive root is a nonzero nonnegative combination of simple roots, so $(\rho ,\beta )>0$ for positive $\beta$, and the pairing is nonzero for every root. In particular $\beta (V_{\rho })=i\beta (H_{\rho })=i(\beta ,\rho )\mathrm{\ne }0$.

Let $S=\overline{\exp (\mathbb{R}{V}_{\rho })}\subseteq T$. Continuity of multiplication and inversion makes this a closed subgroup; it is abelian, compact and connected as the closure of a connected subgroup, hence a torus by [L9]. Naturality implies that $a$ centralizes its dense one-parameter subgroup, and therefore $S$. The closed subgroup $C_G(S)$ has Lie algebra consisting of the vectors fixed by every $\mathrm{Ad}(s)$: necessity follows by differentiating conjugation, and sufficiency by naturality of the exponential and its local charts. The root decomposition in [L1] and $\beta (V_{\rho })\mathrm{\ne }0$ show that this fixed algebra is exactly $\mathfrak{t}$. By [L4] the centralizer is connected. It contains $T$, and the exponential charts of these two groups with the same Lie algebra show $T$ is open in it; connectedness gives $C_G(S)=T$. Thus $a\in T$. If the root set is empty, [L1] gives $\mathfrak{t}=\mathfrak{g}$, $\rho =V_{\rho }=0$ and $S=\{e\}$; the same open-subgroup argument gives $G=T$ and both Weyl groups are trivial.

Hence $g\in n^{\mathrm{\prime }-1}T$, so $w\in w^{\mathrm{\prime }-1}W(\mathrm{\Phi })\subseteq W(\mathrm{\Phi })$; combined with step 3.2, the analytic Weyl group $W(G,T)$ equals the root-system Weyl group $W(\mathrm{\Phi })$, and the cocharacters ${\alpha }^{\vee }$ of step 2.1 supply the claimed reflections with pairing $⟨\alpha ,{\alpha }^{\vee }⟩=2$. The Axiom of Choice entered only through the cited metric, structure and integration theory.