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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.

  1. How can one compute the vector bundle explicitly in the example?
  2. More generally, is there a way to compute the rank of $p_*q^*L$?
  3. How can we compute the weights associated to $p_*q^*L$?
  4. 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.

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    $\begingroup$ Do you know the BBW statement for projections between different flag varieties of a group $G$? $\endgroup$
    – Sasha
    Jul 14, 2022 at 17:45
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    $\begingroup$ Just to add one more comment to the suggestion of @Sasha : the stabilizer group $P/(P\cap Q)$ is itself of the form $H/R$ for a Levi factor $H$ of $P$ and a maximal parabolic $R$ in $H$. So now you can apply Borel-Weil-Bott to work out the $H$-representation of the global sections of the invertible sheaf on $H/R$. Then you can "twist" this representation to get the vector bundle on $G/P$. $\endgroup$ Jul 14, 2022 at 19:54
  • $\begingroup$ @Sasha: I know the statement about the dimension of the cohomology of the line bundles being related to dominant weights associated to the root system of $G$. Maybe it is trivial, but how can one pass to the vector bundles? $\endgroup$
    – Bobech
    Jul 15, 2022 at 6:54
  • $\begingroup$ @JasonStarr: Can you be more precise about this "twist"? I know that $H/R$ is contained in some $\mathbb P(V)$, where $V$ is a $H$-representation. Is my vector bundle related to $G \times^P V$? $\endgroup$
    – Bobech
    Jul 15, 2022 at 6:56
  • $\begingroup$ @Bobech: This is the absolute version of BBW for line bundles, but there is also a relative version, and this is what you need here. $\endgroup$
    – Sasha
    Jul 15, 2022 at 7:40

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