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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.

flag	asked Feb 1 2010 at 10:56 unknown (google) 121●4

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 2010 at 13:42

George McNinch · Answer 1 · 2010-03-22 19:39:02Z UTC

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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).

answered Mar 22 2010 at 19:39

George McNinch
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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 2010 at 20:26

Kevin McGerty · Answer 2 · 2010-02-01 15:13:03Z UTC

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.

Marty · Answer 3 · 2010-02-01 12:35:09Z UTC

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..

Proof of Borel-Weil-Bott Theorem

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