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

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?

• Just in case you didn't already know this: Proposition 6.3 of Kostant (ams.org/mathscinet-getitem?mr=142696) recasts Bott's theorem in a way that bypasses relative Lie algebra cohomology. (See especially loc. cit., equation (6.3.1). Kostant then shows how this reduces the computation of $H^*(G/P,\mathcal{E})$ to his Theorem 5.14.) – Francois Ziegler Jun 28 '13 at 17:42
• As Francois points out, Kostant's Annals paper (available online via JSTOR) is a basic reference. Aside from that, I'd urge you to edit your notation to make the distinction between Lie groups and Lie algebras precise. Your formulation is confusing. – Jim Humphreys Jun 28 '13 at 18:46

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.)