Algebraic root subgroups from root exponentials

By [F1] fix a faithful finite-dimensional rational representation $\rho :G\to GL(V)$ whose comorphism is surjective and identify $G$ with the closed subgroup scheme $\rho (G)\subseteq GL(V)$; then $\mathfrak{g}$ is a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ and $d{\rho }_e$ is the inclusion $\mathfrak{g}\hookrightarrow \mathfrak{g}\mathfrak{l}(V)$, so we may regard $e_{\alpha }\in \mathfrak{g}\subseteq \mathfrak{g}\mathfrak{l}(V)$ as an endomorphism of the finite-dimensional space $V$.

The subalgebra of $\mathfrak{g}$ spanned by $e_{\alpha },f_{\alpha },h_{\alpha }$ is isomorphic to ${\mathfrak{s}\mathfrak{l}}_2$ with standard basis $(e,f,h)$ by [F2], so restriction makes $V$ a finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$-module; by [F4] it is a direct sum of irreducible submodules, and by [F5] the operator $h_{\alpha }$ acts diagonalisably with integer eigenvalues on $V$ and $e_{\alpha }$ raises each eigenvalue by $2$, while on an irreducible submodule the eigenvalue set is $m,m-2,\dots ,-m$. Since $V$ is a finite direct sum of such modules and the eigenvalues occurring are therefore bounded above and below, some positive power of $e_{\alpha }$ annihilates $V$, that is $e_{\alpha }$ is a nilpotent endomorphism of $V$.

Because $e_{\alpha }$ is nilpotent, say $e_{\alpha }^N=0$, the series $\exp (z{e}_{\alpha })={\sum }_{k\ge 0}{z}^k{e}_{\alpha }^k\mathrm{/}k!$ is a finite sum, so each matrix entry of $\exp (z{e}_{\alpha })$ is a polynomial in $z$: the map $z\mapsto \exp (z{e}_{\alpha })$ is a morphism of varieties ${\mathbb{A}}^1\to GL(V)$, and $\exp (z{e}_{\alpha })$ is a unipotent matrix for every $z\in \mathbb{C}$.

The closed-immersion representation $\rho :G\to GL(V)$ of step 1.1 is a homomorphism of finite-dimensional real Lie groups with $d{\rho }_e$ the inclusion $\mathfrak{g}\hookrightarrow \mathfrak{g}\mathfrak{l}(V)$, so [F6] applied to $X=z{e}_{\alpha }\in \mathfrak{g}$ gives $\rho ({\exp }_G(z{e}_{\alpha }))=\exp (z{e}_{\alpha })$ for every real $z$; identifying $G$ with its image in $GL(V)$, this says that $\exp (z{e}_{\alpha })\in G$ for every real $z$.

Choose polynomial functions $f_1,\dots ,f_m$ generating the vanishing ideal of the closed subvariety $G\subseteq GL(V)$. Each composite $z\mapsto f_i(\exp (z{e}_{\alpha }))$ is a polynomial in $z$ by step 2.1 and vanishes for every real $z$ by step 2.2, hence is the zero polynomial; so $\exp (z{e}_{\alpha })\in G$ for every $z\in \mathbb{C}$, and $u_{\alpha }:{\mathbb{A}}^1\to G$, $u_{\alpha }(z)=\exp (z{e}_{\alpha })$, is a well-defined morphism of varieties.

The morphism $u_{\alpha }$ is a group homomorphism: since $[e_{\alpha },e_{\alpha }]=0$ the commuting elements $z{e}_{\alpha }$ and $w{e}_{\alpha }$ satisfy $\exp ((z+w)e_{\alpha })=\exp (z{e}_{\alpha })\exp (w{e}_{\alpha })$, that is $u_{\alpha }(z+w)=u_{\alpha }(z)u_{\alpha }(w)$, and $u_{\alpha }(0)=1$. Its differential at $0$, computed through the closed embedding of step 1.1, sends the generator $1$ of $\mathrm{Lie}{\mathbb{A}}^1=\mathbb{C}$ to $e_{\alpha }\mathrm{\ne }0$, so $d{u}_{\alpha }$ is injective.

For $t\in T$ the conjugation map $c_t:G\to G$, $c_t(g)=tg{t}^{-1}$, is an automorphism of algebraic groups with $d{c}_t=\mathrm{Ad}(t)$; by [F8] and the definition of the character $\alpha$ in the statement, $\mathrm{Ad}(t)$ acts on ${\mathfrak{g}}_{\alpha }$ as $\alpha (t)$, so [F6] applied to $c_t$ and $X=z{e}_{\alpha }$ gives $t{u}_{\alpha }(z)t^{-1}=c_t(\exp (z{e}_{\alpha }))=\exp (\mathrm{Ad}(t)(z{e}_{\alpha }))=\exp (\alpha (t)z{e}_{\alpha })=u_{\alpha }(\alpha (t)z)$ for all $t\in T$ and $z\in \mathbb{C}$. Hence $T$ normalizes and stabilizes $U_{\alpha }$.

Write $E=d{\rho }_e(e_{\alpha })$, so $E^N=0$ for some $N\ge 2$ by step 1.2 and $E\mathrm{\ne }0$ by faithfulness in step 1.1. Choose a linear functional $\mathrm{\ell }:\mathrm{End}(V)\to \mathbb{C}$ with $\mathrm{\ell }(E)=1$. On all of $G$ define the regular function $r(g)=\mathrm{\ell }({\sum }_{j=1}^{N-1}\frac{(-1{)}^{j+1}}{j}(\rho (g)-I{)}^j)\mathrm{.}$ This is a polynomial in the regular matrix entries of $\rho (g)$. In the nilpotent algebra $\mathbb{C}[E]\mathrm{/}(E^N)$ the finite formal identities $\log (\exp (zE))=zE$ hold, so $r(u_{\alpha }(z))=\mathrm{\ell }(zE)=z$ for every $z$ and, as a polynomial identity, for every test $\mathbb{C}$-algebra. Thus $r\circ u_{\alpha }={\mathrm{id}}_{{\mathbb{A}}^1}$ as morphisms of schemes.

Since $G$ is affine, the morphism $r:G\to {\mathbb{A}}^1$ is separated. Its section $u_{\alpha }$, established in step 4.3, is therefore a closed immersion: the graph of the section is the inverse image of the diagonal of the separated scheme $G$ under $({\mathrm{id}}_G,u_{\alpha }\circ r)$, and its image is exactly the equalizer of these two morphisms. Put $U_{\alpha }=u_{\alpha }({\mathbb{A}}^1)$ with this closed subscheme structure. The restriction $r{\mathrm{\mid }}_{U_{\alpha }}$ is a regular inverse to $u_{\alpha }$, so $u_{\alpha }:{\mathbb{G}}_a\stackrel{\sim }{\to }{U}_{\alpha }$ is an isomorphism of algebraic group schemes, not merely a bijection on complex points. By step 4.1 its differential takes $1$ to $e_{\alpha }$, hence $\mathrm{Lie}{U}_{\alpha }=\mathbb{C}{e}_{\alpha }={\mathfrak{g}}_{\alpha }$ by [F3].

If $e_{\alpha }$ is replaced by $c{e}_{\alpha }$ with $c\in {\mathbb{C}}^{\times }$, then $\exp (zc{e}_{\alpha })=u_{\alpha }(cz)$ by the same exponential series, so the image subgroup $U_{\alpha }$ is unchanged; applying the construction of step 1.1 to the root $-\alpha$ and the vector $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ of [F2] produces the opposite closed subgroup $U_{-\alpha }$ with $\mathrm{Lie}{U}_{-\alpha }={\mathfrak{g}}_{-\alpha }$, and [F2] also gives that $e_{\alpha },f_{\alpha },h_{\alpha }$ span a copy of ${\mathfrak{s}\mathfrak{l}}_2$. The Axiom of Choice is assumed in the statement and supplies the countable-choice hypothesis for exponential naturality [F6] at steps 2.2 and 4.2; the finitely many choices of $\rho$, $\mathrm{\ell }$, $f_1,\dots ,f_m$ and $f_{\alpha }$ add no choice principle.