Calogero-Moser Models. II Symmetries and Foldings

Universal Lax pairs (the root type and the minimal type) are presented for CalogeroMoser models based on simply laced root systems, including E8. They exist with and without spectral parameters and they work for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic. For the elliptic potential, the discrete symmetries of the simply laced models, originating from the automorphism of the extended Dynkin diagrams, are combined with the periodicity of the potential to derive a class of CalogeroMoser models known as the ‘twisted non-simply laced models’. For untwisted non-simply laced models, two kinds of root type Lax pairs (based on long roots and short roots) are derived which contain independent coupling constants for the long and short roots. The BCn model contains three independent couplings, for the long, middle and short roots. The G2 model based on long roots exhibits a new feature which deserves further study.