In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:



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> A representation *Ri* of a group *G* should be seen as a quantum object. This representation should be obtained by quantizing a classical theory. The Borel-Weil-Bott theorem gives a canonical way to exhibit for every representation *R* of a compact group *G* a problem in classical physics, with *G* symmetry, such that the quantization of this classical problem gives back *R* as the quantum Hilbert space. One introduces the "flag manifold" *G/T*, with *T* being a maximal torus in *G*, and for each representation *R* one introduces a symplectic structure ω*R* on *G/T*, such that the quantization of the classical phase space *G/T*, with the symplectic structure ω*R*, gives back the representation *R*. **Many aspects of representation theory find natural explanations by thus regarding representations of groups as quantum objects that are obtained by quantization of classical physics.** [page 372; emphasis added]
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I'm fascinated by this idea — I haven't seen it before, but it seems natural, in that classical objects should not be linear, whereas quantum objects should be. I'm most interested in the last sentence: what examples can y'all come up with of representation-theoretic facts that can be "explained" by "physics" on *G/T*? (Besides, of course, Witten's application in the paper I quoted from.)


More generally, I've read the Wikipedia discussion of the Borel-Weil-Bott theorem, and done some random googling, but I haven't found an elementary description of the symplectic structure Witten refers to. Anyone want to pedantically spell out Witten's comment, please?



Examples of applications of the Borel-Weil-Bott theorem?
Look at the orbits of `G` on `g*` the dual of the Lie algebra. These 'coadjoint orbits' have a canonical symplectic structure. Each of these orbits intersects the positive Weyl chamber exactly once; consider those intersecting it at a 'positive weight' (i.e. the the elements of `g*` that lift to characters on `T`, the maximal torus). The positive weights exactly classify the irreducible representations. At the same time, we can build a line bundle over the corresponding coadjoint orbit so that the symplectic form realises the Chern class. Sections of this bundle are automatically a representation of G, and the Borel-Weil-Bott theorem, in one of its friendlier guises, says that this is the representation you expect -- the one with the highest weight we started with.


Generically, a coadjoint orbit is just G/T. When it isn't, it's a deeper quotient, but you can pull back the canonical symplectic structure to G/T, and still do everything there. This is the symplectic structure ωR Witten is referring to.


