Rank-one SL2 homomorphism and Weyl representative

Identify $⟨e_{\alpha },f_{\alpha },h_{\alpha }⟩$ with ${\mathfrak{s}\mathfrak{l}}_2$ by $e\mapsto e_{\alpha }$, $f\mapsto f_{\alpha }$, $h\mapsto h_{\alpha }$ using [F1]; this is a Lie algebra isomorphism onto its image, so the resulting inclusion $\varphi :{\mathfrak{s}\mathfrak{l}}_2\to \mathfrak{g}$ is injective.

The group $S{L}_2(\mathbb{C})$ is connected and simply connected: it is connected as an irreducible algebraic variety, and the map $g\mapsto S$ of the polar decomposition [F5] retracts $S{L}_2(\mathbb{C})$ onto $S{U}_2$ by $g_t=S{\sqrt{g^{\ast }g}}^t$, which is continuous in $(g,t)$ and fixes $S{U}_2$; the determinant-one positive-definite factors are contractible by the path $U^t=\exp (t\log U)$ (the Hermitian logarithm has trace zero), so the inclusion $S{U}_2\hookrightarrow S{L}_2(\mathbb{C})$ is a homotopy equivalence. The parametrisation $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)\mapsto (a,b)$ identifies $S{U}_2$ with the unit sphere $S^3\subseteq {\mathbb{C}}^2\cong {\mathbb{R}}^4$, which is simply connected by [F6]; hence $S{L}_2(\mathbb{C})$ is simply connected.

By [F4] applied to the real Lie groups $S{L}_2(\mathbb{C})$ and $G(\mathbb{C})$ (whose Lie algebras are ${\mathfrak{s}\mathfrak{l}}_2$ and $\mathfrak{g}$ as real Lie algebras) and the homomorphism $\varphi$ of step 1.1, there is a unique smooth homomorphism $\mathrm{\Phi }:S{L}_2(\mathbb{C})\to G(\mathbb{C})$ with $d{\mathrm{\Phi }}_e=\varphi$; independently, applying [F4] to ${\mathfrak{s}\mathfrak{l}}_2\to \mathfrak{g}\mathfrak{l}(V)$ along a faithful representation $G\hookrightarrow GL(V)$ of [F8] shows that $\mathrm{\Phi }$ is the restriction of the corresponding linear integration, hence holomorphic.

First prove regularity on the diagonal, rather than using it in a Gauss chart before it exists. Choose the faithful rational closed immersion $\rho :G\hookrightarrow GL(V)$ of [F8]. Restrict $d\rho \circ \varphi$ to the ${\mathfrak{s}\mathfrak{l}}_2$-module $V$ and decompose $V={⨁}_{m\in \mathbb{Z}}{V}_m$ into $h$-eigenspaces by [F9]. For $u=\exp (z)\in {\mathbb{C}}^{\times }$, exponential naturality [F7] gives $\rho \mathrm{\Phi }(\mathrm{diag}(u,u^{-1})){\mathrm{\mid }}_{V_m}=\exp (zm)\mathrm{id}=u^m\mathrm{id}$; the integer exponents make this independent of the logarithm of $u$. Thus in a weight basis the matrix entries of $\rho \mathrm{\Phi }{\mathrm{\mid }}_D$ are Laurent monomials, so $\mathrm{\Phi }{\mathrm{\mid }}_D$ is an algebraic morphism because $G$ is a closed subscheme of $GL(V)$. On $d\mathrm{\ne }0$ the Gauss decomposition is $g=u_{+}(b\mathrm{/}d)\mathrm{diag}(1\mathrm{/}d,d)u_{-}(c\mathrm{/}d)$; on $a\mathrm{\ne }0$ it is $g=u_{-}(c\mathrm{/}a)\mathrm{diag}(a,a^{-1})u_{+}(b\mathrm{/}a)$. On the remaining chart $b\mathrm{\ne }0$ one has $g=u_{-}(d\mathrm{/}b)w\mathrm{diag}(-1\mathrm{/}b,-b)u_{-}(a\mathrm{/}b)$, as direct matrix multiplication using $ad-bc=1$ verifies. These three principal opens cover $S{L}_2$. By [F2], [F7] and the diagonal regularity, the expression for $\mathrm{\Phi }$ on each chart is a product of algebraic morphisms and the fixed point $\mathrm{\Phi }(w)$; their agreement follows from the already defined smooth homomorphism $\mathrm{\Phi }$. Hence $\mathrm{\Phi }$ is a morphism of algebraic groups, denoted ${\phi }_{\alpha }$.

Item (i) is step 2.1, and item (ii) follows: for $X=e$ the exponential series gives $\mathrm{\Phi }\left(\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right)\right)=\mathrm{\Phi }({\exp }_{{\mathfrak{s}\mathfrak{l}}_2}(ze))={\exp }_G(z{e}_{\alpha })=u_{\alpha }(z)$ by [F2], [F7], and similarly for the transpose with $f$ and $f_{\alpha }$; for $g\in \ker \mathrm{\Phi }$, differentiating $\mathrm{\Phi }(gx{g}^{-1})=\mathrm{\Phi }(x)$ at $x=1$ gives $d{\mathrm{\Phi }}_e(\mathrm{Ad}(g)X)=d{\mathrm{\Phi }}_e(X)$ for every $X\in {\mathfrak{s}\mathfrak{l}}_2$; since $d{\mathrm{\Phi }}_e$ is injective by step 1.1, $\mathrm{Ad}(g)$ fixes every $X\in {\mathfrak{s}\mathfrak{l}}_2$; thus $g\in \ker \mathrm{Ad}=\{\pm I\}$, the last equality being the standard centre of $S{L}_2(\mathbb{C})$, so $\ker \mathrm{\Phi }\subseteq \{\pm I\}$.

Item (iii) is a definition plus one computation: ${\alpha }^{\vee }=\mathrm{\Phi }{\mathrm{\mid }}_D$ is a morphism of algebraic groups into $T$: the complex exponential map $z\mapsto \mathrm{diag}(e^z,e^{-z})$ is surjective onto $D$, and exponential naturality [F7] for the inclusion $T\hookrightarrow G$ shows $\mathrm{\Phi }(\mathrm{diag}(e^z,e^{-z}))={\exp }_G(z{h}_{\alpha })={\exp }_T(z{h}_{\alpha })\in T$. Its differential satisfies $d{\alpha }_1^{\vee }(1)=d{\mathrm{\Phi }}_e(h)=h_{\alpha }$, since the curve $u\mapsto diag(u,u^{-1})$ has derivative $h$ at $1$; for a character $\lambda \in X^{\ast }(T)$ the composite $\lambda \circ {\alpha }^{\vee }:{\mathbb{C}}^{\times }\to {\mathbb{C}}^{\times }$ is a morphism of algebraic groups, hence of the form $u\mapsto u^m$ for a unique integer $m$, and differentiating at $u=1$ gives $m=d\lambda (h_{\alpha })$, where $d\lambda :\mathfrak{h}\to \mathbb{C}$ is the differential of the character; writing $⟨\lambda ,{\alpha }^{\vee }⟩=\lambda (h_{\alpha })=d\lambda (h_{\alpha })$ gives $\lambda ({\alpha }^{\vee }(u))=u^{⟨\lambda ,{\alpha }^{\vee }⟩}$.

In $S{L}_2$ the stated Weyl matrix factors as $w=u_{+}(-1)u_{-}(1)u_{+}(-1)$, as direct multiplication shows. Hence $n_{\alpha }={\phi }_{\alpha }(w)=u_{\alpha }(-1)u_{-\alpha }(1)u_{\alpha }(-1)$. For $H\in \mathfrak{h}$ write $H=H_0+\frac{\alpha (H)}{2}{h}_{\alpha }$, where $[H_0,e_{\alpha }]=[H_0,f_{\alpha }]=0$ by the sl2 relations of [F1]. Therefore all three root-subgroup factors centralize $H_0$. The matrix $w$ conjugates $h$ to $-h$ in ${\mathfrak{s}\mathfrak{l}}_2$, so $n_{\alpha }$ sends $h_{\alpha }$ to $-h_{\alpha }$ under the integrated homomorphism. Consequently $\mathrm{Ad}(n_{\alpha })(H)=H_0-\frac{\alpha (H)}{2}{h}_{\alpha }=H-\alpha (H)h_{\alpha }=s_{\alpha }(H)$.

Hence $n_{\alpha }\in N_G(T)$: $\mathrm{Ad}(n_{\alpha })$ preserves $\mathfrak{h}$ by step 4.1, so conjugation by $n_{\alpha }$ maps the closed connected subgroup $T$ to a closed connected subgroup with Lie algebra $\mathfrak{h}$ and the same dimension, which must be $T$ itself; moreover $\mathrm{Ad}(n_{\alpha }){\mathrm{\mid }}_{\mathfrak{h}}=s_{\alpha }$ is the reflection of the root system on $\mathfrak{h}$, corresponding dually to the reflection $s_{\alpha }$ on $X^{\ast }(T)$, so $\lambda (n_{\alpha }t{n}_{\alpha }^{-1})=(s_{\alpha }\lambda )(t)$ for every character $\lambda$. Finally $w^2=-I$ gives $n_{\alpha }^2={\phi }_{\alpha }(-I)={\exp }_G(\pi h_{\alpha })\in T$, an element of $T$ acting trivially on $T$, so the square of the Weyl representative is central in $T$ rather than a new condition.

Collecting the preceding steps gives the morphism ${\phi }_{\alpha }$ of the statement with the differential of (i), the root subgroup identifications and kernel bound of (ii), the coroot and character pairing of (iii) and the Weyl representative of (iv). The Axiom of Choice enters through [F4] and [F7], whose countable-choice interfaces are inherited from AC, and through the published root and highest-weight suppliers behind [F1] and [F9]; [F8] is explicitly choice-free; the only selections made in the argument are the fixed $e_{\alpha },f_{\alpha }$ and the finite data of the three affine charts.