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

A representation

Rof a group_{i}Gshould 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 representationRof a compact groupGa problem in classical physics, withGsymmetry, such that the quantization of this classical problem gives backRas the quantum Hilbert space. One introduces the "flag manifold"G/T, withTbeing a maximal torus inG, and for each representationRone introduces a symplectic structure ω_{R}onG/T, such that the quantization of the classical phase spaceG/T, with the symplectic structure ω_{R}, gives back the representationR.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?

explainrepresentations to people who don't know them? Well, one way would be to take a group of rotations of our world (which is deep down a 2-cover of`SO(1;3)`

) as say: hey, a representation is a linear object that knows how to transform under this. Which is, essentially, a definition of a particle in quantum mechanics. $\endgroup$ – Ilya Nikokoshev Nov 5 '09 at 20:26Theoretical Physics StackExchange. [1]: theoreticalphysics.stackexchange.com/questions/551/… $\endgroup$ – John Sidles Nov 21 '11 at 22:16