Generalized Flag Variety

In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties.

Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.

The most general concept of a generalized flag variety is a conjugacy class of parabolic subgroups of a semisimple algebraic or Lie group G: G acts transitively on such a conjugacy class by conjugation, and the stabilizer of a parabolic P is P itself, so that the generalized flag variety is isomorphic to G/P. It may also be realised as the orbit of a highest weight space in a projectivized representation of G. In the algebraic setting, generalized flag varieties are precisely the homogeneous spaces for G which are complete as algebraic varieties. In the smooth setting, generalized flag manifolds are the compact flat model spaces for Cartan geometries of parabolic type, and are homogeneous Riemannian manifolds under any maximal compact subgroup of G.

Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces.