Reductive Groups

A reductive group is the central object of the structure theory: a connected linear algebraic group whose geometric unipotent radical is trivial. Reductive groups encompass tori, semisimple groups, and their canonical extensions, and they are the groups appearing on the "automorphic side" of the Langlands programme.

Definition

Let \(G\) be a connected smooth affine algebraic group over a field \(k\), and let \(R(G)\), \(R_u(G)\) denote the radical (= maximal connected solvable normal subgroup) and unipotent radical (= maximal connected unipotent normal subgroup).

\(G\) is reductive if the geometric unipotent radical \(R_u(G_{\bar k}) = 1\). Equivalently, \(G_{\bar k}\) has no nontrivial connected unipotent normal subgroup.

\(G\) is semisimple if \(R(G_{\bar k}) = 1\) — strictly stronger than reductive.

Geometric vs.\ algebraic

Over a non-perfect field \(k\) the geometric condition (after base change to \(\bar k\)) is genuinely stronger than the bare condition "\(R_u(G) = 1\)". For \(k\) perfect (in particular for char \(0\) or finite \(k\)), the two coincide.

Structure theorems

For a connected reductive group \(G\) over an algebraically closed field \(k\):

Radical decomposition. \(G = Z(G)^\circ \cdot G_{\mathrm{der}}\), with \(Z(G)^\circ\) a torus (the connected centre) and \(G_{\mathrm{der}} = [G, G]\) semisimple. The intersection \(Z(G)^\circ \cap G_{\mathrm{der}}\) is a finite group of multiplicative type.
Borel subgroups exist and are conjugate.
Maximal tori exist and are conjugate.
Levi decomposition of parabolics. Every parabolic \(P \subseteq G\) is a semidirect product \(P = L \ltimes R_u(P)\) with \(L\) a reductive Levi.

Classification

Theorem (Chevalley, Demazure-Grothendieck). Split connected reductive groups over any field \(k\) are classified up to isomorphism by their based root datum. Over \(\mathbb{Z}\), each based root datum defines a Chevalley group scheme.

Non-split forms are classified by Galois cohomology of the quasi-split form (see node).

Examples

Tori. \(\mathbb{G}_m\), \(\mathbb{G}_m^n\), norm tori \(R^{(1)}_{K/k}\mathbb{G}_m\), Weil restrictions \(R_{K/k}\mathbb{G}_m\).
Classical groups. \(\mathrm{GL}_n\), \(\mathrm{SL}_n\), \(\mathrm{Sp}_{2n}\), \(\mathrm{SO}_n\), \(\mathrm{U}_n\) (unitary for a Hermitian form), and their inner forms over global fields.
Exceptional groups. \(G_2, F_4, E_6, E_7, E_8\) in simply connected, adjoint, and intermediate isogeny forms; their inner forms.
Quasi-split forms. Outer forms determined by the action of \(\mathrm{Gal}(\bar k/k)\) on the Dynkin diagram (e.g.\ \({}^2 A_n\), \({}^3 D_4\), \({}^2 E_6\)).

Non-examples

\(\mathbb{G}_a\) is connected, smooth, affine, but \(R_u\) = itself, hence not reductive.
The Heisenberg group is unipotent, not reductive.
A semidirect product of \(\mathbb{G}_m\) and \(\mathbb{G}_a\) (the "\(ax + b\)" group) has \(R_u = \mathbb{G}_a\), hence not reductive.

Role in Langlands

Reductive groups are the natural setting for:

Automorphic forms. For \(G\) reductive over a global field \(F\), the automorphic representations of \(G(\mathbb{A}_F)\).
Local Langlands. For \(G\) reductive over a local field \(F\), the parametrisation of representations by L-parameters into \({}^L G\).
Galois representations. For \(G = \mathrm{GL}_n\), the Langlands correspondence with \(n\)-dimensional Galois representations.

Mathlib status

Mathlib.AlgebraicGeometry.GroupScheme.* provides group schemes abstractly.
Reductive groups specifically are not yet formalised in Mathlib; this project's affine algebraic groups node provides the starting point.

References

Borel, Linear Algebraic Groups (2nd ed., 1991).
Springer, Linear Algebraic Groups (2nd ed., 1998).
Milne, Reductive Groups (notes), §17–§25.
Conrad, Reductive group schemes (SGA3 summer school notes, 2014).
SGA 3, vol. III (Demazure–Grothendieck, 1970).