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Post Made Community Wiki by S. Carnahan
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Integrable system theory might be an example, however the examples I can propose requires some technicality level (which probably undesired)...

The line of application is the following

Questions (seemingly not related to Lie groups): Consider the differential operator - $H = \sum \partial_i^2 + \sum exp(x_i - x_{i+1} )$

Qeust 1: Can you find some differential operators which commute with H? Quest 2: Can you find eigenvectors for H ?

(This is called Toda quantum system, Calogero and some other can be considered as well)

Lie groups comes into the game like this:

The main idea is that this differential operator comes from the Casimir of gl_n, so higher Casimirs (i.e. Z(U(gl_n)) will provide the commiting operators. What we need to do to obtain this operator from standard Casimir - is to make some reduction (integration) over some some subgroup. Eigenfucntions can be obtained by intgration of some characters of the representations, which comes from the fact that Casimirs acts by scalars on any irreps...

The sl(2) example is simple technically and for me it was quite a beatiful, when it was explained to me... In sl(2) case we get Bessel functions as eigenfunctions, some properties like recurrent formulas for different "n" in Bessel can be derived from tensor decomposition of corresponding representations...

If necessary I can provide refrences...

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These are "quantum integrable" systems, similar can be done for classical ones - one can obtain solutions of diffurs.