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The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of projective space. See N. Chriss & V. Ginzburg 's book "Representation Theory and Complex Geometry" section 5.7.

Now let $G$ be a simple algebraic group over $\mathbb{C}$, $B$ and $T$ be Borel subgroup and maximal torus contains in $B$, respectively. Let $\mathcal{B}=G/B$ be completed flag variety. Let $\mathcal{O}(\alpha)$ be line bundle over $\mathcal{B}$ parameterized by integral weight($\alpha\in Hom(T,\mathbb{C}^*)$). Also denote it by $\Delta$ the diagonal embedding of $\mathcal{B}$ to $\mathcal{B}\times\mathcal{B}$. My question is: how to give a locally free sheaves resolution of the sheaf $\Delta_*\mathcal{O}(\alpha)$? I hope there is a similiar resolution like Beilinson resolution. Any suggestion is welcome ,thanks!

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    $\begingroup$ Have you seen the following paper? Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92 (1988), no. 3, 479--508. $\endgroup$
    – Sasha
    Commented Mar 12 at 10:07
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    $\begingroup$ @Sasha, Thanks for your suggestion ! I have seen Kapranov's paper just now. In his paper, he considered the special case of flag variety, i.e. take G=GL(V)(or SL(V)). But I hope a resolution for arbitrary simple type of algebraic group G, maybe his result can be extended to the another type. Thanks for your suggestion again! $\endgroup$
    – fool rabbit
    Commented Mar 12 at 11:04
  • $\begingroup$ Beilinson (and Kapranov) construct resolutions by means of exceptional collections; these resolutions are very special. On the other hand, for any coherent sheaf on a smooth projective variety one can construct a locally free resolution just by iteratively covering the sheaf by a sufficiently large multiple of a sufficiently negative line bundle. Is the latter enough for you, or do you need the former? $\endgroup$
    – Sasha
    Commented Mar 12 at 12:33
  • $\begingroup$ @Sasha This seems not enough for me. In fact, I want to do same explicit calculation (e.g. convolution) on the K group of some variety relate to flag variety. Use some technique, it is enough for me to know the class (sum of line bundle over B×B) of direct image sheaf of line bundle (on \mathcal{B} ). Then I can do my calculation. So I need a explicit formulas of such classes, which parameterized by integral weight. So I wonder if a Beilinson type resolution exist for any line bundle over B, then I can determine the class what I need in the K^0(B×B). So I want to a "canonical resolution" $\endgroup$
    – fool rabbit
    Commented Mar 12 at 13:07
  • $\begingroup$ It is conjectured that an exceptional collection exists for any flag variety, see the introduction to [Kuznetsov, Alexander; Polishchuk, Alexander. Exceptional collections on isotropic Grassmannians. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 3, 507--574] for a survey. However, for your purposes a numerically exceptional collection should be enough, and this, I think, can be obtained with a torus localization technique. $\endgroup$
    – Sasha
    Commented Mar 12 at 18:14

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