Topic Classification of Split Reductive Groups
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Classification of Split Reductive Groups

The Chevalley-Demazure-Grothendieck theorem classifies split connected reductive groups by their based root data. This is the foundational theorem of the structure theory, putting the entire zoo of classical and exceptional groups onto a uniform footing.

Statement

Theorem (Chevalley 1955; Demazure-Grothendieck SGA3, 1962-64). Let \(k\) be a field. The map \[ \{\text{split connected reductive } k\text{-groups}\}/\cong \;\longrightarrow\; \{\text{based root data}\}/\cong \] \[ (G, T, B) \;\longmapsto\; (X^*(T), \Phi, X_*(T), \Phi^\vee; \Delta) \] is a bijection between isomorphism classes of split connected reductive groups (with chosen pinning) and isomorphism classes of based root data.

Moreover, every based root datum lifts to a Chevalley group scheme \(G_{\mathbb{Z}}\) defined over \(\mathbb{Z}\), unique up to isomorphism, with \(G = G_{\mathbb{Z}} \times_{\mathbb{Z}} k\) the split form over \(k\).

Pinned form of the theorem

A pinning of a split reductive group \((G, T, B)\) is a choice of non-zero root vector \(X_\alpha \in \mathfrak{g}_\alpha\) for each simple root \(\alpha \in \Delta\). The pinning rigidifies the group: the pinned automorphism group \(\mathrm{Aut}^{\mathrm{pin}}(G, T, B, \{X_\alpha\})\) equals the automorphism group of the based root datum.

Without pinning, one has an exact sequence: \[ 1 \;\to\; G_{\mathrm{ad}}(k) \;\to\; \mathrm{Aut}(G) \;\to\; \mathrm{Out}(G) \;\to\; 1, \] where \(\mathrm{Out}(G) = \mathrm{Aut}(\text{based root datum})\) is the group of diagram automorphisms.

Existence: Chevalley's construction

For each based root datum, Chevalley constructed the corresponding split group via:

  1. Choose a complex semisimple Lie algebra \(\mathfrak{g}_{\mathbb{C}}\) with the given root system.
  2. Choose a Chevalley basis \(\{X_\alpha, H_\alpha\}\) of \(\mathfrak{g}_{\mathbb{C}}\) with integer structure constants.
  3. Define \(G_{\mathbb{Z}}\) as the group scheme generated by the formal "exponentials" \(\exp(t X_\alpha)\) and the maximal torus, subject to the Steinberg relations.

The resulting group scheme works over any commutative ring.

Uniqueness: the isomorphism theorem

Theorem. If \((G, T, B)\) and \((G', T', B')\) are split connected reductive groups over \(k\) with isomorphic based root data \((X, \Phi, X^\vee, \Phi^\vee; \Delta) \cong (X', \Phi', \ldots; \Delta')\) via some isomorphism \(\phi\), then there exists an isomorphism \(f : G \to G'\) inducing \(\phi\) on root data. The choice of \(f\) is unique up to composition with conjugation by \(T(k)\).

Examples of classification

Based root datum Group
\(A_{n-1}\), \(X = \mathbb{Z}^n\) \(\mathrm{GL}_n\)
\(A_{n-1}\), \(X = P\) (weight lattice) \(\mathrm{SL}_n\)
\(A_{n-1}\), \(X = Q\) (root lattice) \(\mathrm{PGL}_n\)
\(B_n\) sc \(\mathrm{Spin}_{2n+1}\)
\(B_n\) ad \(\mathrm{SO}_{2n+1}\)
\(C_n\) sc \(\mathrm{Sp}_{2n}\)
\(D_n\) sc \(\mathrm{Spin}_{2n}\)
\(E_6\) sc \(E_6^{\mathrm{sc}}\) (78-dim Lie group)
\(G_2\) (= sc = ad) the exceptional group \(G_2\)

Working over \(\mathbb{Z}\)

The Chevalley group scheme \(G_{\mathbb{Z}}\) has fibres \(G_{\mathbb{Z}} \times_{\mathbb{Z}} \mathbb{F}_p = G_p\) over each prime, all of which are "the same group in characteristic \(p\)" (modulo bad-prime subtleties). This makes the classification work universally:

  • \(G(\mathbb{F}_q)\): finite group of Lie type, with order given by Steinberg's formula in terms of the root system;
  • \(G(\mathbb{Q}_p)\): \(p\)-adic group of Lie type;
  • \(G(\mathbb{C})\): complex Lie group;
  • \(G(\mathbb{R})\): real Lie group (split form).

Non-split forms

Once the classification of split forms is fixed, non-split forms over a field \(k\) are classified by Galois cohomology of the automorphism group: \[ \{\text{forms of split } G \text{ over } k\} \;\cong\; H^1(k, \mathrm{Aut}(G)). \]

The exact sequence \(1 \to G_{\mathrm{ad}} \to \mathrm{Aut}(G) \to \mathrm{Out}(G) \to 1\) gives a decomposition into inner forms (in \(H^1(k, G_{\mathrm{ad}})\)) and outer forms (Galois-equivariant pinned-automorphism cocycles).

Role in Langlands

  • Langlands functoriality: requires the classification — to speak of \({}^L G\), one needs the dual group \(\hat G\), which is defined by reversing the root datum. The classification is what makes this dual group well-defined and unique.
  • Reductive group schemes (SGA3): the existence of \(G_{\mathbb{Z}}\) enables the global Langlands programme on Shimura varieties at all primes simultaneously.
  • Categorification: the geometric Satake equivalence \(\mathrm{Perv}(\mathrm{Gr}_G) = \mathrm{Rep}(\hat G)\) is a "categorification" of the classification.

Mathlib status

Mathlib.LinearAlgebra.RootSystem.IsCrystallographic and related modules formalise the root-system side; the algebraic-group side (constructing \(G\) from a root datum) is not yet there.

References

  • Chevalley, Sur certains groupes simples (Tôhoku 1955).
  • Demazure–Grothendieck, SGA 3, vol.\ III (1970): the definitive treatment.
  • Steinberg, Lectures on Chevalley Groups (Yale notes, 1968).
  • Conrad, Reductive group schemes (Notes, 2014).
  • Milne, Reductive Groups (notes), §24.

Prerequisite Topics

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