Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times\_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic of $E$ in terms of the inducing representation $V$?


I am also interested in the question of how the $G$-module of holomorphic sections of $E$ relates to $V$, that is, the first cohomology group of $E$.



Euler characteristic of a holomorphic homogeneous vector bundle
In the case $G$ is a complex semisimple Lie group and $P$ its parabolic subgroup, the answer is given by Kostant's version of Borel-Bott-Weil theorem [K]. Any homogeneous vector bundle is given by a associated bundle construction from some $P$-representation $\mathbb{F}$. In general, one can consider also $P$-representations which do not have highest weight $\lambda$, but let's stick to those that do are actually highest weight representations $\mathbb{F}\_\lambda.$ In the case when $\lambda$ is $\mathfrak{g}$-integral and $\mathfrak{g}$-dominant, the cohomology of $E \simeq G\times\_P\mathbb{F}\_\lambda$ is calculated by Lie algebra cohomology of the nilradical $\mathfrak{u}$ of $\mathfrak{p} = \mathfrak{l} \oplus \mathfrak{u}$ and its $k$-th degree is given by
$$\bigoplus\_{w \in W^\mathfrak{l}, l(w) = k} \mathbb{F}\_{w\cdot \lambda}$$
where $W^\mathfrak{l}$ is the subset of minimal coset representatives of $W/W\_{\mathfrak{l}}$ and $w\cdot \lambda = w(\lambda + \rho) - \rho.$


There are some generalization for weights which are not of this type. Look for "Kostant modules"


[K] Kostant, Bertram[Lie algebra cohomology and the generalized Borel-Weil theorem.](https://mathscinet.ams.org/mathscinet-getitem?mr=142696)
Ann. of Math. (2) 74 (1961), 329–387. 


