Liouville Integrability of a Class of Integrable Spin Calogero-Moser Systems and Exponents of Simple Lie Algebras

In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of Chevalley on the Lax operators with spectral parameter. As part of our analysis, we will develop several results concerning the algebra of invariant polynomials on simple Lie algebras and their expansions.