If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type.
According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view.
Relative to this basis, the stabilizer of the standard flag is the group of nonsingular lower triangular matrices, which we denote by Bn.
The complete flag variety can therefore be written as a homogeneous space GL(n,F) / Bn, which shows in particular that it has dimension n(n−1)/2 over F. Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a Borel subgroup.
Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements.
When k=2, this is a Grassmannian of d1-dimensional subspaces of V. This is a homogeneous space for the general linear group G of V over F. To be explicit, take V = Fn so that G = GL(n,F).
First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups.
The presence of a complex structure and cellular (co)homology make it easy to see that the cohomology ring of G/H is concentrated in even degrees, but in fact, something much stronger can be said.
Let us now restrict our coefficient ring to be a field k of characteristic zero, so that, by Hopf's theorem, H*(G) is an exterior algebra on generators of odd degree (the subspace of primitive elements).
It follows that the edge homomorphisms of the spectral sequence must eventually take the space of primitive elements in the left column H*(G) of the page E2 bijectively into the bottom row H*(BH): we know G and H have the same rank, so if the collection of edge homomorphisms were not full rank on the primitive subspace, then the image of the bottom row H*(BH) in the final page H*(G/H) of the sequence would be infinite-dimensional as a k-vector space, which is impossible, for instance by cellular cohomology again, because a compact homogeneous space admits a finite CW structure.
The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map H*(BG) → H*(BH) induced by the inclusion of H in G. The map H*(BG) → H*(BT) is injective, and likewise for H, with image the subring H*(BT)W(G) of elements invariant under the action of the Weyl group, so one finally obtains the concise description where
Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the fundamental class), and indeed, as hoped.
If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety.