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I found on the literature plenty of articles dealing with connections between rational/trigonometric/elliptic Calogero-Moser systems and their relativistic generalizations (Ruijsenaars-Schneider), and other fields on mathematics and physics. There was also a topic discussing with the trichotomy R/T/E here: Groups, quantum groups and (fill in the blank).

I have a really naive question: why the hyperbolic systems do not appear in the story?

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the integrable systems in question are complexified, so for an n-body model (say Ruijsenaars-Schneider) the phase space is $4n$ real ($2n$ complex)-dimensional. If the coordinates and conjugate momenta are complex this covers trigonometric and hyperbolic at once.

If you really want to study hyperbolic as opposed to trigonometric you need to impose certain reality conditions on the spectrum.

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  • $\begingroup$ Thank you. That brings me with another (kinda vague) question, what about quantization? quantum trigonometric RS systems are solved by Macdonalds polynomials, but the hyperbolic ones are much more complicated. Only the rank one is understood, and the Hilbert space is infinite dimensional. Why is it so different? $\endgroup$ – Julien Dec 29 '18 at 6:11
  • $\begingroup$ Formal spectrum of complexified trigonometric RS models are given by q-hypergeometic series. When certain reality condition (trigonometric projection) + quantization are met they truncate to Macdonald polynomials. It does not happen in the hyperbole case. In my recent paper [arXiv:1805.00986] I discuss this truncation and connection of the spectrum of the tRS model to enumerative geometry. $\endgroup$ – Peter Koroteev Dec 30 '18 at 7:04
  • $\begingroup$ So the solution is known for any N. $\endgroup$ – Peter Koroteev Dec 30 '18 at 7:05

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