nLab parabolic subgroup

Definition

Given a linear algebraic group $G$ (i.e. an algebraic subgroup of the general linear group $GL(n,k)$ where $k$ is a field), a subgroup $P\subset G$ is said to be parabolic if it is closed in Zariski topology and the quotient 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.

The Cartan geometry of parabolic subgroup inclusions is parabolic geometry.

Related concepts

• Schubert calculus

• parabolic geometry

References

• eom parabolic subgroup