It is well-known that the set of irreducible (finite-dimensional) representations of a semisimiple complex Lie algebra g can be indexed by the dominant weights. The Borel-Weil theorem asserts that they can be seen geometrically as the global sections of line bundles over the flag variety. The Borel-Weil-Bott theorem computes the higher sheaf cohomology groups. There are several ways to prove the Borel-Weil-Bott theorem, which we will discuss. The classical idea is to study how the Casimir operator acts on the sheaf of sections of line bundles. Instead of this, the geometric idea is trying to compute the Doubeault cohomology, transferring the sheaf cohomology to the Lie algebra cohomology. The algebraic idea is to realize that the sheaf cohomology group can be computed by the derived functor of the induction, by using the Peter-Weyl the Borel-Weil theorem can be shown immediately.
