Split reductive algebraic group

A connected reductive algebraic group over a field $k$ is a smooth connected affine $k$ -group $G$ whose geometric unipotent radical is trivial. It is split over $k$ if it contains a maximal torus

T\simeq\mathbb G_m^r

split over $k$ .

Root datum over the base field

For split $G$ , the character and cocharacter lattices of $T$ , roots, coroots, and a choice of Borel subgroup are defined without extending the base field. A split connected reductive group is classified up to isomorphism by its root datum.

Examples include $\operatorname{GL}_n$ , $\operatorname{SL}_n$ , and $\operatorname{Sp}_{2n}$ in their standard forms over $k$ .

Integral models

A split root datum defines a Chevalley–Demazure reductive group scheme over $\mathbb Z$ . Over a nonarchimedean local field this provides a hyperspecial subgroup after base change to the valuation ring. For a nonsplit but unramified group, a reductive integral model can still exist even though the group is not split over the local field itself.

Duality and descent

For split $G$ , the Galois action on the based root datum is trivial, so the $L$ -group has trivial Weil action on $\widehat G$ . For a nonsplit group, the dual group can be the same abstract complex group while the Weil action differs; this extra action is essential data.

Relation to the letter

The letter begins with split root data and then applies Galois descent and inner twisting. This cleanly separates the combinatorial dual group from the arithmetic $F$ -form.

References

Michel Demazure and Alexander Grothendieck, eds., Schémas en groupes (SGA 3).
A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.