Rank-One Subgroups and SL₂-Embeddings

For each root \(\alpha \in \Phi(G,T)\) of a connected reductive group, there is a canonical rank-one subgroup \(G_\alpha \subseteq G\) through which the algebraic group \(\mathrm{SL}_2\) (or \(\mathrm{PGL}_2\)) embeds, carrying the standard root structure into the root system of \(G\).

The rank-one subgroup \(G_\alpha\)

Define \[ G_\alpha = \langle U_\alpha, U_{-\alpha} \rangle \subseteq G, \] the closed subgroup generated by the root subgroup \(U_\alpha \cong \mathbb{G}_a\) and its opposite \(U_{-\alpha} \cong \mathbb{G}_a\).

Theorem. \(G_\alpha\) is a connected semisimple group of rank one, and there is a surjective group scheme morphism \[ \varphi_\alpha : \mathrm{SL}_2 \longrightarrow G_\alpha \] with central kernel \(\mu_2\) (if \(\alpha\) is not divisible in \(X^*(T)\)) or trivial kernel. Under \(\varphi_\alpha\): \[ \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \mapsto x_\alpha(t), \quad \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} \mapsto x_{-\alpha}(t), \quad \begin{pmatrix} u & 0 \\ 0 & u^{-1} \end{pmatrix} \mapsto \alpha^\vee(u). \]

The coroot as a cocharacter

The coroot \(\alpha^\vee \in X_*(T) = \mathrm{Hom}(\mathbb{G}_m, T)\) is exactly the cocharacter obtained as the composite \[ \mathbb{G}_m \hookrightarrow \mathrm{SL}_2 \xrightarrow{\varphi_\alpha} G, \] where the first map sends \(u \mapsto \begin{pmatrix} u & \\ & u^{-1} \end{pmatrix}\).

This gives a canonical identification: \[ \alpha^\vee = (\varphi_\alpha)_*\begin{pmatrix} 1 & \\ & -1 \end{pmatrix} \in X_*(T). \]

Weyl group elements from rank-one subgroups

For each root \(\alpha\), the element \[ n_\alpha = x_\alpha(1)\, x_{-\alpha}(-1)\, x_\alpha(1) = \varphi_\alpha \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \in N_G(T)(k) \] represents the simple reflection \(s_\alpha \in W(G,T) = N_G(T)/T\). These Weyl group representatives satisfy the braid relations and generate \(N_G(T)\) over \(T\).

SL₂ vs PGL₂

Whether the morphism \(\varphi_\alpha\) factors through \(\mathrm{PGL}_2\) or only \(\mathrm{SL}_2\) depends on the isogeny type of \(G\) and the root \(\alpha\):

• If \(\alpha^\vee\) maps \(-1 \in \mathbb{G}_m\) to a non-trivial element of \(G\), then the kernel is trivial and \(\varphi_\alpha\) is an embedding of \(\mathrm{SL}_2\) (isogeny of degree 1 onto \(G_\alpha\)).
• If \(\alpha^\vee(-1) = 1\), then \(\varphi_\alpha\) factors through \(\mathrm{PGL}_2\).

For the simply-connected form, \(\mathrm{SL}_2 \to G_\alpha\) is always an isomorphism; for the adjoint form, \(\mathrm{PGL}_2 \to G_\alpha\) is always an isomorphism.

Relation to sl₂-triples

At the Lie algebra level, the differential of \(\varphi_\alpha\) sends the standard \(\mathfrak{sl}_2\)-triple \(\{e, f, h\}\) to the triple \(\{E_\alpha, E_{-\alpha}, H_\alpha\}\) in \(\mathrm{Lie}(G)\), which is an \(\mathfrak{sl}_2\)-triple in the sense that \[ [H_\alpha, E_\alpha] = 2 E_\alpha, \quad [H_\alpha, E_{-\alpha}] = -2 E_{-\alpha}, \quad [E_\alpha, E_{-\alpha}] = H_\alpha. \] This is the connection between the algebraic-group and Lie-algebra theories of root systems.

Uniqueness of \(\varphi_\alpha\)

The morphism \(\varphi_\alpha\) is not unique: it depends on the choice of root subgroup parametrizations \(x_\alpha, x_{-\alpha}\). Changing \(x_\alpha(t) \mapsto x_\alpha(c t)\) for \(c \in k^\times\) changes \(\varphi_\alpha\) by a \(T(k)\)-conjugation. A pinning fixes a canonical choice of \(x_\alpha\) for each simple root, making \(\varphi_\alpha\) canonical for simple roots.