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I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which is an algebra generated by Dunkl operators and some group elements. (see Lecture notes on Cherednik Algebras)

I am curious about the physical aspects of this algebra/operators, since I have heard that Dunkl operators arise naturally in the Hamiltonian mechanics of quantum systems. As someone not very well versed in physics (and since every resource I was able to find is research level), I would appreciate a "simpler" explanation.

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Well, this all has definitely something to do with 2D CFT. Actually, Knizhnik-Zamolodchikov equations are mentioned on the page 43 of the link you provided. If you're not familiar with this topic, di Francesco is the Bible, Blumenhagen is a short intro.

Besides this, it's been long noticed that equations similar to 2D CFT arise in the context of 1D quasi-exactly solvable (QES) QM models. Here, in Section 8, the Calogero model is discussed in quite detail. All these topics are quite intriguing from the mathematical perspective. We probably still don't have a deep understanding of relations between QES QM models and 2D CFT.

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