Homogeneous vector bundles

Let $G$ be a connected complex semisimple Lie group, and $P$ a parabolic subgroup. Then $G$ is a principal $P$ -bundle over the homogeneous manifold $X= G/P$ . Any holomorphic representation $\phi : P \to \operatorname{Aut}(E)$ on a complex vector space $E$ induces a holomorphic vector bundle $\mathbb{E}$ over $X$ :

$$\mathbb{E}:= G \times_{\phi} E := (G \times E)/ \sim,$$

where $(gp,e)\sim (g, \phi(p) e)$ . Then $\mathbb{E}$ is a holomorphic vector bundle over $X= G/P$ . A vector bundle over $X$ arising in this way is called a homogeneous vector bundle. Let $\mathcal{O}(\mathbb{E})$ be the corresponding sheaf of holomorphic sections. The homogeneous vector bundle $\mathbb{E}$ inherits a left $G$ -action from the left multiplication in $G$ :

$$h.(g,e)=(hg, e)$$   for $$h,g \in G, e \in E.$$

Thus, all the cohomology groups $H^q(X, \mathcal{O}(\mathbb{E}))$ become $G$ -modules.

In [15] Bott proved that if the representation $\phi$ is irreducible, the cohomology groups $H^q(X, \mathcal{O}(\mathbb{E}))$ all vanish except possibly in one single dimension. Moreover, in the nonvanishing dimension $q$ , $H^q(X, \mathcal{O}(\mathbb{E}))$ is an irreducible representation of $G$ whose highest weight is related to $\phi$ .

This theorem generalizes an earlier theorem of Borel and Weil, who proved it for a positive line bundle.

In Bott's paper one finds a precise way of determining the nonvanishing dimension in terms of the roots and weights of $G$ and $P$ . Thus, on the one hand, Bott's theorem gives a geometric realization of induced representations, and on the other hand, it provides an extremely useful vanishing criterion for the cohomology of homogeneous vector bundles.