Is there a generalization of Borel-Weil-Bott theorem for not completely reducible vector bundles?

Question

Let G be simple algebraic group, P its parabolic subgroup. Then the category of G-equivariant vector bundles on G/P is equivalent to the category of representations of P. Borel-Weil-Bott tells us how to compute cohomology of the bundles corresponding to irreducible representations of P, and so also of the bundles corresponding to completely reducible representations of P. But some important G-equivariant bundles on G/P are not completely reducible, for example the tangent bundle of $\operatorname{SO}(2n+1)/P_i$, where $P_i$ is the i-th maximal parabolic subgroup of $\operatorname{SO}(2n+1)$. So my question is: is there a way to compute cohomology of such bundles? Maybe there exists some class of groups including parabolic subgroups of reductive groups and some class of representations of such groups which is larger than completely reducible representations but still manageable?

Answer 1

I'm not aware of any generalization that is strong enough to compute the cohomology of such bundles completely, but you can at least use the Borel--Weil--Bott theorem to get some vanishing results. This was already done by Bott in his 1957 Annals paper to show that the cohomology of the tangent bundle $T$ of $G/P$ vanishes in degree > 0. The first step of this computation is to note that the action of $P$ on the tangent space at $P/P$ can be identified, as a $P$-module, with $\mathfrak g/ \mathfrak p$ (with $P$ acting via Ad). Consequently, the short exact sequence of $P$-modules $$ 0 \to \mathfrak p \to \mathfrak g \to \mathfrak g / \mathfrak p \to 0 $$ yields a short exact sequence of $G$-equivariant vector bundles $$ 0 \to G \times_P \ \mathfrak p \to G \times_P \ \mathfrak g \to T \to 0 $$ on $G/P$. The middle bundle is trivial, and so we get an isomorphism $H^q(G/P, T) \cong H^{q+1}(G/P, G \times_P \ \mathfrak p)$ for all $q>0$.

It remains to compute $H^\ast(G/P, G \times_P \ \mathfrak p)$. For this, a vanishing result of the Borel--Weil--Bott type is useful. This result states that if you have a $G$-equivariant vector bundle $\mathcal V = G \times_P V$ on $G/P$ then $H^q(G/P, \mathcal V) = 0$ unless $q$ is equal to the index of some nonsingular $\mu + \rho$, where $\mu$ is one of the highest weights of $V$.

For the details, you can either try Bott's original paper, or checkout section 11 of this paper of Dennis Snow. Another paper of potential interest is

P.A. Griffiths, Some geometric and analytic properties of homogeneous complex manifolds. I. Sheaves and cohomology, Acta Math. 110 (1963), 115–155.

Answer 2

I have dealt with this situation many times, and as far as I can tell, you just need to chase some long exact sequences (or spectral sequences if you want).

There is a paper by Ottaviani and Rubei http://arxiv.org/abs/math/0403307 that translates the problem of calculating cohomology of a homogeneous bundle (when the base space is a Hermitian symmetric variety) into doing some calculations over some quiver with relations. I was never able to actually use this to get an answer, but it may be interesting for you.