Flag manifold

Summary

A flag manifold is a quotient $G/P$ of a connected complex semisimple Lie group $G$ by a parabolic subgroup $P$. The basic example is the space of complete flags $V_1 \subset V_2 \subset \cdots \subset V_{n-1}$ of subspaces of $\C^n$ with $\dim V_i = i$. Projective spaces and Grassmannians are flag manifolds too. Flag manifolds are compact complex manifolds on which $G$ acts, and they carry the geometry behind the Borel–Weil theorem and Schubert calculus.