nLab
parabolic subgroup

Given a linear algebraic group $G$ (i.e. an algebraic subgroup of $\mathrm{GL}(n,k)$ where $k$ is a field), a subgroup $P\subset G$ is said to be parabolic if it is closed in Zariski topology and the question variety $G/P$ is projective. A minimal (with respect to inclusion) parabolic subgroup of a linear algebraic group is called a Borel subgroup; in fact, given a Borel subgroup $B$, any closed subgroup $P\supset B$ is parabolic.

• eom parabolic subgroup