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Is there any purely algebraic proof of Borel-Weil-Bott theorem. I mean only techniques from Algebraic group. In each and every proof I have seen so far they use Lie group techniques and then translate to Algebraic group version. I need a proper reference which is easily readable.

Thanks in advance.

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There is a proof by Lunts-Rosenberg on quantum analogue of Borel-weil-Bott theorem using purely algebraic or algebraic geometry(categorical geometry)way. Check out the paper: localization for quantum group –  Shizhuo Zhang Mar 10 '10 at 13:42
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3 Answers 3

The simplest proof of Borel-Weil-Bott that I know is due to Demazure: he has two papers in Inventiones (one in 1968 the other in 1976) on the theorem, and the second is two pages long -- it gives a simplification of his previous proof, and he uses only algebro-geometric techniques. Both papers are readable.

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Near the bottom of Jacob Lurie's homepage, you can find an exposition of the Borel-Weil-Bott theorem from an algebro-geometric standpoint. It is "easily readable" if you're familiar with the things like line bundles on projective varieties, semisimple algebraic groups, etc..

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See J.C. Jantzen "Representations of Algebraic Groups" II.5 especially II.5.5.

There you will find an algebraic proof of the result in char. 0 (probably more-or-less Demazure's proof, mentioned in another answer). And you'll find a proof of some of what remains true in positive char (due to Henning Anderson).

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Demazure's approach is certainly the most useful from the viewpoint of algebraic geometry in characteristic 0, but the related questions in characteristic $p>0$ remain to a large extent open and are natural follow-ups. As George indicates, Jantzen's book provides access to such questions in a unified framework. Andersen got started on his own work partly by exploring how Demazure's set-up might be adapted to characteristic $p$. But a full analogue of Bott's theorem probably requires some creative use of Kazhdan-Lusztig theory for the affine Weyl group. –  Jim Humphreys Mar 22 '10 at 20:26
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