TITLE: Rep Theory Consequences of Bott--Weil--Borel
QUESTION [28 upvotes]: 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?

REPLY [6 votes]: 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.)