1) It is quite common in mathematics that a beautiful theory appears when we can understand an interesting example from a new perspective which captures its essence:
In our case, the interesting example is the Kepler problem of planetary motions. Just as Lebesgue's theory put Riemann integrals and series on an equal footing, our theory put the harmonic oscillators and the Kepler problem on an equal footing in the sense that an isotropic harmonic oscillator is equivalent to the bound state sector of a generalized Kepler problem. However, our theory offers a bit more: while a harmonic oscillator has no scattering sector, its equivalent formulation does.
2) It is a fact that a lot of interesting mathematics has the "physics" side, and the "physics" side is often geometric and often provides refined information:
In our case, the mathematics is the (nontrivial) unitary lowest weight representations for simple real non-compact Lie groups, and the theory we are developing here is the "physics" side. The "physics" side offers much more because the physics models have many aspects, such as the classical dynamic problem, quantum scattering problem, duality, etc.