Generalized Flag Variety - Symmetric Spaces

Symmetric Spaces

Let G be a semisimple Lie group with maximal compact subgroup K. Then K acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety G/P is a compact homogeneous Riemannian manifold K/(K∩P) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous Kähler manifold.

Turning this around, the Riemannian homogeneous spaces

M = K/(K∩P)

admit a strictly larger Lie group of transformations, namely G. Specializing to the case that M is a symmetric space, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano.

If G is a complex Lie group, the symmetric spaces M arising in this way are the compact Hermitian symmetric spaces: K is the isometry group, and G is the biholomorphism group of M.

Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under K are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking G to be a real form of the biholomorphism group Gc of a Hermitian symmetric space Gc/Pc such that P := Pc∩G is a parabolic subgroup of G. Examples include projective spaces (with G the group of projective transformations) and spheres (with G the group of conformal transformations).