Chevalley Group Construction from a Root Datum

Given a based root datum \(\Psi = (X, \Phi, \Delta, X^\vee, \Phi^\vee, \Delta^\vee)\), or equivalently a pinning \((G, B, T, (e_\alpha)_{\alpha \in \Delta})\) over an algebraically closed field, the Chevalley–Demazure construction builds a canonical split reductive group scheme \(\mathbf{G}_\Psi\) over \(\mathbb{Z}\).

Step 1: The semisimple Lie algebra and Chevalley basis

Starting from the abstract root datum, choose the corresponding semisimple Lie algebra \(\mathfrak{g}\) over \(\mathbb{Q}\) with root system \(\Phi \subset X \otimes \mathbb{R}\). Chevalley (1955) showed that \(\mathfrak{g}\) admits a \(\mathbb{Z}\)-basis \[ \{H_\alpha : \alpha \in \Delta\} \cup \{e_\alpha : \alpha \in \Phi\} \] with structure constants in \(\mathbb{Z}\): \[ [H_\alpha, e_\beta] = \langle \beta, \alpha^\vee \rangle\, e_\beta,\quad [e_\alpha, e_{-\alpha}] = H_\alpha,\quad [e_\alpha, e_\beta] = N_{\alpha,\beta}\, e_{\alpha+\beta} \] where \(N_{\alpha,\beta} \in \mathbb{Z}\) depend only on the root system and can be chosen with a sign convention (the Chevalley basis).

The \(\mathbb{Z}\)-lattice \(\mathfrak{g}_\mathbb{Z} = \operatorname{span}_\mathbb{Z}\{H_\alpha, e_\alpha\}\) is a Lie algebra over \(\mathbb{Z}\) and is the Chevalley \(\mathbb{Z}\)-form.

Step 2: Root subgroups \(U_\alpha\)

For each root \(\alpha \in \Phi\), define the root subgroup by the exponential of the nilpotent \(e_\alpha\)-action. In the construction, one introduces elements \[ x_\alpha(t) = \exp(t\, e_\alpha) = \sum_{k=0}^\infty \frac{t^k}{k!} e_\alpha^k \] acting on a representation. For a suitable faithful representation \((\rho, V)\) with \(\mathbb{Z}\)-lattice \(L \subset V\), \(\rho(e_\alpha)^k/k!\) maps \(L\) to \(L\) for all \(k\), so \(x_\alpha(t)\) is a well-defined element of \(\mathrm{GL}(L)\) for any \(t\) in a \(\mathbb{Z}\)-algebra \(R\).

The map \(t \mapsto x_\alpha(t)\) defines a group scheme homomorphism \[ x_\alpha : \mathbb{G}_a \longrightarrow \mathrm{GL}(L_R) \] whose image \(U_\alpha \cong \mathbb{G}_a\) is the root subgroup for \(\alpha\). The key point is that this is defined over \(\mathbb{Z}\).

Step 3: The torus \(T\)

The character lattice of the maximal torus is the group \(X = X^*(T)\) specified by the root datum. Define \[ T = \operatorname{Hom}_\mathbb{Z}(X, \mathbb{G}_m) = \operatorname{Spec}(\mathbb{Z}[X]) \] where \(\mathbb{Z}[X]\) is the group ring of \(X\). This is a split algebraic torus over \(\mathbb{Z}\) with \(X^*(T) = X\).

Step 4: Chevalley commutator relations

The root subgroups and torus satisfy the following relations (Chevalley commutator formulas):

1. Torus action on root subgroup: \[ t \cdot x_\alpha(c) \cdot t^{-1} = x_\alpha(\alpha(t)\, c) \] for \(t \in T(R)\), \(c \in R\), where \(\alpha(t) = \chi_\alpha(t) \in R^\times\) is the character \(\alpha : T \to \mathbb{G}_m\).

2. Commutator of root subgroups: For \(\alpha, \beta \in \Phi\) with \(\alpha + \beta \ne 0\), \[ [x_\alpha(s), x_\beta(t)] = \prod_{i,j > 0,\, i\alpha+j\beta \in \Phi} x_{i\alpha+j\beta}(C_{\alpha\beta ij}\, s^i t^j) \] where the product is over roots of the form \(i\alpha+j\beta\) and \(C_{\alpha\beta ij} \in \mathbb{Z}\).

3. Inverse and Weyl elements: For each simple root \(\alpha\), define \[ n_\alpha(t) = x_\alpha(t)\, x_{-\alpha}(-t^{-1})\, x_\alpha(t),\quad h_\alpha(t) = n_\alpha(t)\, n_\alpha(-1)^{-1}. \] Then \(h_\alpha : \mathbb{G}_m \to T\) is the coroot \(\alpha^\vee\) viewed as a cocharacter, and \(n_\alpha(1)\) represents the simple reflection \(s_\alpha \in W\).

Step 5: The group \(\mathbf{G}_\Psi\)

The Chevalley group scheme \(\mathbf{G}_\Psi\) over \(\mathbb{Z}\) is defined as the closed subgroup scheme of \(\mathrm{GL}(L)\) generated by all \(U_\alpha\) and \(T\). More precisely: \[ \mathbf{G}_\Psi = \text{(group scheme generated by } T, \{U_\alpha : \alpha \in \Phi\}). \] By the Bruhat decomposition (which holds at the level of functors on \(\mathbb{Z}\)-algebras), this group scheme is affine of finite type over \(\mathbb{Z}\), smooth and flat, with geometrically connected fibres.

Properties of the output

Independence of choices

The construction depends on:

• The choice of Chevalley basis (sign conventions for structure constants \(N_{\alpha,\beta}\)): different choices give isomorphic groups.
• The choice of representation \(V\): different representations give isomorphic groups (by the Chevalley isomorphism theorem).

The resulting group scheme \(\mathbf{G}_\Psi\) is therefore canonical up to isomorphism given the based root datum \(\Psi\).

Examples

(SC = simply connected; adj = adjoint.)

Relation to the Chevalley classification theorems

• Existence (Chevalley existence theorem): Every root datum arises from some connected reductive group. The construction above provides the explicit witness.
• Isomorphism (Chevalley isomorphism theorem): Any two split connected reductive groups with the same based root datum are uniquely isomorphic as pinned groups. The construction above gives the canonical representative.

The two theorems together give the bijection: \[ \{\text{split connected reductive groups over } k\}/\cong \;\xrightarrow{\sim}\; \{\text{based root data}\}/\cong. \]