For a given flag manifold, the stabilisers of different points are all conjugate to each other. In addition, all parabolic subgroups contain a conjugate of the subgroup of upper triangular matrices. These are basic objects in algebraic geometry. What follows is an elegant example of the interactions between representation theory, complex analysis, and the underlying geometry of flag manifolds. In this case the subgroup is B , the upper triangular matrices.
Holomorphic maps play an important role and the process is sometimes called holomorphic induction .
It can be extended trivially to all of B since the strictly upper triangular matrices are nilpotent. The Borel-Weil theorem says the subspace of holomorphic equivariant functions is irreducible and is the dual of the irreducible representation corresponding to the weight we started with. Bott extended this theorem to parabolic subgroups other than B . Otherwise there is only one non-trivial cohomology group and it realises an irreducible representation. Homogeneous Vector Bundles , Ann.