I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory of complex geometry. 


What I can't seem to find out, though, is if this fact has been used to prove any interesting results in representation theory, ie, does the geometric realization of the representation allow us to prove anything we didn't know already? 



Rep Theory Consequences of Bott--Weil--Borel
It's always a good idea to ask (as students typically do) why one is studying a particular subject or theorem. Here are some of my views, from the algebraic side of representation theory:


1) The original theorem here was proved by Borel and Weil, though never written up formally by them. Serre reported on it at the Bourbaki seminar (expose 100, usually available online at NUMDAM, which may be under renovation at the moment). Taken by itself, the Borel-Weil theorem provides a somewhat concrete geometric model using line bundles on the flag variety for all finite dimensional irreducible representations of a (complex or compact) semisimple Lie group. Up to isomorphism these representations are parametrized by "dominant" characters of a maximal torus. The existence was originally an indirect consequence of work by E. Cartan and then Weyl, but the actual representations are not easy to write down. Instead, some indirect information about characters (or weight space multiplicities) was cleverly developed. 


2) Later Bott, in his fundamental 1957 *Annals* paper "Homogenous vector bundles" responded to a conjecture of Borel and Hirzebruch by proving that for non-dominant weights and corresponding line bundles, the flag variety has non-vanishing cohomology in at most one (predictable) degree. When the cohomology is non-zero, it then affords the same irreducible representation you get from a domiannt weight "linked" by the Weyl group via Borel-Weil. From the viewpoint of representation theory, this of course is a somewhat negative result showing that nothing new turns up in higher cohomology calculations.


3) In a series of papers (mostly in *Invent. Math.* and available online via GDZ archive), Demazure translated these ideas into the language of algebraic geometry. Still working in characteristic 0, he derived a "very simple" proof of the theorems of Borel-Weil and Bott, then showed how to derive the Weyl character formula and implement it effectively in this same framework.


4) As pointed out by Aakumadule, Kumar's proof of the old PRV conjecture on tensor products of irreducibles (predicting certain direct summands) relies heavily on the machinery of the Borel-Weil theorem, though in the algebraic framework. Naturally the flag variety is a major player in representation theory here and elsewhere, as in Kumar's work with Littelmann and the book by Brion and Kumar on Frobenius splitting. This all tends to drift into prime characteristic too.


5) In prime characteristic (which interests me more), work by H.H. Andersen and others has shown what can and can't be done in Demazure's set-up. In particula, the reductions modulo a prime of the standard irreducible representations become "Weyl modules" (and play a major role in Jantzen's book *Representations of Algebraic Groups*). These are usually reducible, but have formal properties like the infinite dimensional Verma modules in characteristic 0 and lead (by Lusztig's ideas) to a partly proved conjecture on characters of irreducibles. For higher cohomology there are still many open problems. Unlike Bott's case, one sometimes has systematic occurrences of multiple non-zero higher cohomology groups though the Euler character remains invariant. I've conjectured that the Kazhdan-Lusztig theory for an affine Weyl group (relative to the rpime) controls all of this in a nice way. 


