In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:
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 ωR on G/T, such that the quantization of the classical phase space G/T, with the symplectic structure ωR, 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]
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?