Lie Algebra Cohomology and the Borel-Weil-Bott Theorem

We have seen that irreducible finite dimensional representations of a complex simple Lie algebra g or corresponding compact Lie group are classified and can be constructed starting from an integral dominant weight. The dominance condition depends upon a choice of positive roots (or equivalently, a choice of invariant complex structure on the flag manifold.) An obvious question is that of what happens if we make a different choice of positive roots, or start with a non-dominant highest weight. The Weyl group permutes possible choices of positive roots, at the same time permuting highest weights. It turns out that there is a generalization of the Borel-Weil theorem which describes the effect of these Weyl group permutations. This is the Borel-WeilBott theorem, which realizes representations in other cohomology degrees, not just the degree-zero case of holomorphic sections. This phenomenon is best understood in terms of the Lie algebra cohomology of the nilpotent radical subalgebra n ⊂ g.