Reductive algebraic group

Definition

Let $k$ be a field. A reductive algebraic group over $k$ is a smooth, connected, affine algebraic group $G$ whose geometric unipotent radical is trivial:

R_u(G_{\overline k})=1.

Passing to an algebraic closure in this definition makes reductivity a geometric property. A connected semisimple group is reductive, as is a torus; $GL_n$ , $SL_n$ , $PGL_n$ , and $Sp_{2n}$ are standard examples.

Role of Borel and parabolic subgroups

After base change to an algebraic closure, $G$ has Borel subgroups. Its parabolic subgroups give projective homogeneous quotients $G/P$ .

References

Brian Conrad, “Reductive group schemes,” in Autour des schémas en groupes, Panoramas et Synthèses 42–43 (2014), 93–444.