Definition Split Reductive Groups
definition admitted

Split Reductive Groups

A connected reductive group \(G\) over a field \(k\) is split if it contains a maximal torus \(T\) that is \(k\)-split, i.e., isomorphic to a product of copies of \(\mathbb{G}_m\) as a \(k\)-group scheme: \[ T \cong \mathbb{G}_m^r \quad \text{over } k, \] where \(r = \mathrm{rank}(G)\).

Split torus

A torus \(T\) is split over \(k\) if its character group \(X^*(T) = \mathrm{Hom}_{k\text{-gp}}(T, \mathbb{G}_m)\) is a free \(\mathbb{Z}\)-module on which \(\mathrm{Gal}(\bar k/k)\) acts trivially. Equivalently, \(T\) is diagonalizable over \(k\) itself (not just over \(\bar k\)).

Split maximal torus

A split maximal torus is a maximal torus among the tori of \(G\) that is split over \(k\). Such a torus need not exist for a general reductive group over a non-algebraically closed field \(k\).

Example. For \(G = \mathrm{SL}_2\) over \(\mathbb{R}\), the diagonal torus \(T = \left\{\begin{pmatrix} t & \\ & t^{-1} \end{pmatrix}\right\}\) is split, while the "compact" torus \(\mathrm{SO}(2)\) is not. So \(\mathrm{SL}_2/\mathbb{R}\) is split.

Example. The unitary group \(\mathrm{U}(n)\) over \(\mathbb{R}\) associated to the standard Hermitian form is not split; its maximal \(\mathbb{R}\)-split torus has rank \(0\).

Classification

The fundamental theorem of Chevalley–Demazure–Grothendieck states that over any field \(k\), split connected reductive groups are classified up to isomorphism by their based root data: \[ \{\text{split connected reductive groups over } k\}/{\cong} \;\xrightarrow{\sim}\; \{\text{based root data}\}/{\cong}. \] This is the content of the Chevalley existence and isomorphism theorems.

Split forms of reductive groups

Over a general field \(k\), a reductive group \(G\) may not be split, but it always has a unique split inner form (the quasi-split or split form), which serves as the canonical representative of the \(k\)-forms of \(G\). All \(k\)-forms of a fixed split group \(G_0\) are classified by the Galois cohomology set \(H^1(k, \mathrm{Aut}(G_0))\).

Over algebraically closed fields

Every reductive group over an algebraically closed field is split: since all tori split over \(\bar k\), and a maximal torus over \(\bar k\) is rational by the conjugacy of maximal tori, the group is automatically split. Thus the Chevalley classification over \(\bar k\) classifies all reductive groups.

Role in the theory

Split reductive groups are the starting point for:

  • Chevalley group construction: building \(G\) from a based root datum.
  • Galois descent: non-split forms are torsors for split groups.
  • Langlands programme: the L-group \({}^L G = \hat G \rtimes W_k\) involves the dual group \(\hat G\), defined using the dual root datum of the split form.

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