Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the associated complex line bundle $G\times\_P V\rightarrow G/P$. Let $X\subseteq G\times\_P V$ denote the complement of the zero-section. I would like to compute $H^\*(X;\mathbb{C})$ in terms of the given data. 


My first attempt has involved identifying $X$ up to homotopy with a circle bundle over $G/P$. The Gysin sequence is then applicable. However, it becomes necessary to understand the maps $$H^i(G/P)\xrightarrow{\cup e} H^{i+2}(G/P),$$ where $e\in H^2(G/P)$ is the Euler class of the circle bundle. If one fixes a maximal torus and Borel $T\subseteq B\subseteq P$, then I think the Euler class equals the weight of the $T$-rep $V$ (where we are identifying $H^2(G/P)$ with $H^2(G/B)^{W\_P}=Hom(T,\mathbb{C}^\*)^{W\_P}$). 


I would appreciate any advice concerning my approach. If you like, you may take $G$ to be simple, and you may take the $T$-weight to be the highest root.


I would also appreciate any alternatives to my approach.


Fundamentally, though, my question is: How should one use the Hard Lefschetz Theorem to understand the maps $H^i(G/P)\xrightarrow{\cup e} H^{i+2}(G/P)$? I would really appreciate a reference on this. 



The Gysin Sequence for an Associated Bundle over a Partial Flag Variety
Here is my comment as an answer.


An excellent reference / example for these ideas is Juteau's paper "Cohomology of the minimal nilpotent orbit" (<http://arxiv.org/abs/0704.3417>). The minimal nilpotent orbit can be described as the the complement of the zero section in a line bundle on $G/P$, where $P$ is the standard parabolic subgroup corresponding to the simple roots orthogonal to the highest root. Hence calculating its cohomology is an example of your question.


Juteau is interested in doing this calculation integrally. This is quite subtle, as this paper shows. Rationally it is much simpler (by the hard Lefschetz theorem as you remark). Over $\mathbb{Q}$ the first half of the cohomology is canonically isomorphic to the primitive part of the cohomology. See Remark 2.4 in Juteau's paper.


