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?

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    $\begingroup$ You can take a look at Shrawan Kumar's paper on the PRV conjecture, which was a conjecture about how to decompose the tensor product of two irreducible finite dimensional representations of a complex simple group$G$. The proof heavily uses the geometry of $G/B\times G/B$ the flag variety of $G\times G$. (Shravan Kumar, Invent math ,1988, vol 93, 117-130. There is also the standard monomial theory, a collection of results in representation theory, proved using geometric methods. $\endgroup$ – Venkataramana Feb 6 '13 at 16:35
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    $\begingroup$ First, I don't think that representations are coming from nowhere. When a group $G$ acts on a space $X$, and you have a $G$-equivariant bundle on the space $X$, then you get a representation of $G$ on the sections (and higher cohomology) of the bundle. Maybe the most impressive results beyond the theorem itself are how useful it is for generalizations. The generalization that comes first to my mind is Schmid's "L²-cohomology and the discrete series" (Annals, 1976) which proved a conjecture of Langlands by using a geometric realization in the spirit of Borel-Weil-Bott. $\endgroup$ – Marty Feb 6 '13 at 18:19
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    $\begingroup$ I think one can regard the BWB theorem as a precursor of the Beilinson Bernstein localization theorem. The latter gives an equivalence between representations of the Lie algebra and D-modules on the flag variety. I think one can hardly overestimate the importance of the latter theorem. It allows to use a lot of geometric machinery to solve representation theoretic problems. For example it was a key ingredient in the original proof of the Kazhdan-Lusztig conjectures. $\endgroup$ – Jan Weidner Feb 7 '13 at 2:06

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.

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    $\begingroup$ This comment by Prof.Humphreys is most informative. I forgot to add in my earlier comment that Habousch's proof of Mumford's conjecture on geometric reductivity of reductive groups" also uses the explicit model afforded by the Borel-Weil theorem (in prime characteristic). $\endgroup$ – Venkataramana Feb 6 '13 at 23:53

What I'm writing here seems more like a contribution to a big-list than an "answer", but since you've already chosen one anyway...

Say you're interested in which irreps $V_\nu$ occur in $V_\lambda \otimes V_\mu$. Let $C$ be the set of triples $(\lambda,\mu,\nu)$ for which this is true, a subset of $($the dominant weights$)^3$. Theorem: $C$ is closed under addition.

The only proof I know is this: $(\lambda,\mu,\nu)$ is in $C$ iff the line bundle ${\mathcal L}_\lambda \boxtimes {\mathcal L}_\mu \boxtimes {\mathcal L}_\nu^*$ over $(G/B)^3$ has a nonzero $G$-invariant section. Given two nonzero sections for two different triples, we can tensor them together, and the result will still be nonzero because $(G/B)^3$ is reduced and irreducible. Hence the sum of the triples is again in $C$.

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Check out the book Frobenius Splitting Methods in Representation Theory by Brion and Kumar. There are lots of representation-theoretic results in positive characteristic in that book that rely crucially on the geometric realization of induced representations. For many of the representation-theoretic facts proved by Frobenius splitting methods it is true that one can also write down a non-geometric proof, but the Frobenius splitting proofs are almost always simpler and case-free (see for example chapter 4 of Brion-Kumar regarding the good filtration property for modules over a reductive group).

And even if you're only interested in complex representations rather than positive-characteristic representations, this technique is still useful because one can base-change facts from positive characteristic to characteristic 0. There is at least one purely representation-theoretic statement over the complex numbers (regarding the so-called generalized Brylinski-Kostant filtration) that I know only a geometric proof of, using Frobenius splitting methods in positive characteristic and then base-changing.

One thing that somewhat complicates this whole picture is that often when a representation-theoretic result is first proved by geometric means it is natural to then look for an algebraic proof, so you'll often find proofs that come in both flavors. (See for example R. Brylinski's original paper on the BK-filtration, which crucially uses geometry to prove a purely representation-theoretic fact; her paper was followed by a paper of Joseph and Heckenberger which gives an alternate algebraic proof of her results. Another example is the algebraic Frobenius splitting technique devised by Kumar and Littelmann.)

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