We work over the field of complex numbers $\mathbb C$.
Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. It is well-known that also $P \cap Q$ is a parabolic subgroup of $G$. Moreover, the homogeneous $G$-variety $G/P \cap Q$ dominates the generalized Grassmannians $G/P$ and $G/Q$. $\require{AMScd}$ \begin{CD} G/P \cap Q @>{q}>> G/Q\\ @V{p}VV\\ G/P. \end{CD} Let $L$ be the generator of the Picard group $\operatorname{Pic}(G/Q) \simeq \mathbb ZL$. I want to describe the vector bundle $p_*q^*L$ over $G/P$ as $P$ varies in the standard maximal parabolic subgroups of $G$.
Example. Let $G \simeq \operatorname{SL}_{n+1}$ and $Q$ be the standard parabolic subgroup associated to the simple root $\alpha_1$ in the root system of the Lie algebra $\mathfrak g \simeq \mathfrak{sl}_{n+1}$, while $P$ is the standard parabolic subgroup associated to the simple root $\alpha_k$ for $1<k\le n$. Then $G/Q \simeq \mathbb P^n$ and $G/P \simeq \operatorname{Gras}(k,n+1)$, the Grassmannian of $k$-planes in a $(n+1)$-vector space. Then $G/P \cap Q \simeq \operatorname{Flag}(1,k;n+1)$ is the flag of lines and $k$-planes in the vector space: \begin{CD} \operatorname{Flag}(1,k;n+1) @>{q}>> \mathbb P^n\\ @V{p}VV\\ \operatorname{Gras}(k,n+1). \end{CD} If $L = \mathcal O(1)$ is the dual of the tautological line bundle over the projective space, then it is known that $p_*q^*L$ is the tautological subbundle $S$ appearing in the universal sequence $$ 0 \to S \to \mathcal O^{\oplus (n+1)} \to Q \to 0. $$
Questions.
- How can one compute the vector bundle explicitly in the example?
- More generally, is there a way to compute the rank of $p_*q^*L$?
- How can we compute the weights associated to $p_*q^*L$?
- Once one knows the weights, how can one reconstruct the associated bundle?
I know that this is related to the Borel-Weil-Bott theorem. Everybody seems to know how to compute the bundles, but I cannot find anywhere a reference to learn it.