On the geometry of the Calogero-Moser system

We discuss a special eigenstate of the quantized periodic Calogero—Moser system associated to a root system. This state has the property that its eigenfunctions, when regarded as multivalued functions on the space of regular conjugacy classes in the corresponding semisimple complex Lie group, transform under monodromy according to the complex reflection representation of the affine Hecke algebra. We show that this endows the space of conjugacy classes in question with a projective structure. For a certain parameter range this projective structure underlies a complex hyperbolic structure. If in addition a Schwarz type of integrality condition is satisfied, then it even has the structure of a ball quotient minus a Heegner divisor. For example, the case of the root system E₈ with the triflection monodromy representation describes a special eigenstate for the system of 12 unordered points on the projective line under a particular constraint.