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.

regulardominant weight then it's a hyperplane class, and Hard Lefschetz will tell you it's injective up to middle dimension. The highest root usually isn't $P$-regular. Probably you'd want to think about fibering $G/P$ over $G/P'$ where the highest root is $P'$-regular, i.e. in the interior of the wall $(T^*_+)^{W_P}$ of the Weyl chamber. – Allen Knutson Sep 7 '13 at 16:01