Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)

Question

Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in Bott's 1957 paper, says that $H^*(G/P,{\cal E}) = \sum K\otimes H^*(p,v,{\rm Hom}(K,E))$, where $p$ is the Lie algebra of $P$, $v$ the Lie algebra of the intersection of $P$ with the maximal compact subgroup $M$ of $G$, and the sum is over all irreducible representations $K$ of $M$.

My question is how to compute the relative Lie algebra cohomology appearing on the RHS of this formulae in practice, say when $M$ is of ADE type (and $G$ its complexification). I understand that in the degree $0$ case, ${\rm H}^0$ is computed simply as homomorphisms from $K$ to $E$ over $p$. What is an efficient way to compute the higher cohomology groups?

Also: the more commonly seen version of the theorem deals with line bundle ($E$ being a $1$-dimensional representation of $P$). In this case, the RHS is usually expressed in terms of the highest weight representation given by a Weyl group transformation of the weight vector associated with that $1$-dimensional representation. How does this result follow from the general formula above expressed in terms of relative Lie algebra cohomology, and in particular, why does the length of the Weyl group element translate into the degree of the cohomology group?

Answer 1

The theorems of Borel-Weil and then Bott, along with Kostant's translation of the ideas into the language of Lie algebra cohomology, do much to illuminate classical representation theory (Cartan-Weyl) but probably can't be viewed as a computational tool. As in other situations, cohomological language provides a natural setting for concrete older ideas and also suggests new possibilities (for instance in the work of Schmid on infinite dimensional Lie group representations). But in Lie theory you don't usually get easier ways to deal with the combinatorics.

While Bott approached the subject rather indirectly, the idea of Borel and Weil is fairly direct and geometric: realize irreducible finite dimensional representations of compact (or complex) semisimple Lie groups in terms of the cohomology of associated line bundles relative to the flag variety. For more general vector bundles, as in Bott's paper, you can expect less explicit results but perhaps more flexibility in the methods.

By using the then-recent spectral sequence methods of Grothendieck in algebraic geometry, Demazure was able to streamline the original Borel-Weil-Bott treatments (working over an algebraically closed field of characteristic 0). He wrote a short paper (in French) here and then a much shorter version of the main step (in English) here.

In all these papers the notation tends to differ a lot, but once you get into the spirit of Demazure's short proof you may be able to see more clearly how the lengths of Weyl group elements correlate with the possible degrees of nonvanishing cohomology groups for arbitrary line bundles. Basically the argument goes step-by-step inductively, starting with a single simple reflection to get from a weight in the dominant Weyl chamber to a weight in an adjacent chamber where the same irreducible representation typically occurs but with a new non-dominant weight attached. (Here the Weyl group reflections occur with origin shifted to $-\rho$, since a canonical line bundle is also involved.)

Again I'd emphasize that no new representations turn up (unless you venture into prime characteristic), but Bott's theorem and by implication the Lie algebra interpretation may seem more natural. Since the combinatorial description of these representations is already intricate (in terms of weight spaces and characters), one has to expect things to get much more complicated to compute for arbitrary vector bundles. But at least there is a coherent pattern.