On the Calogero-Moser solution by root-type Lax pair

Abstract

The “root type Lax pair” for the rational Calogero-Moser system for any simply laced root system yields a solution not for the path q(t) but for the values of the inner products (α, q(t)), where α ranges over all roots of the root system. It does not, however, tell us which value of the inner product corresponds to which root. In the present paper, we show that the solution is indeed uniquely determined by these values (up to root system automorphisms) at almost all times. We show by counterexample that it is possible for two different values of q to yield the same set of values for the inner products (α, q). The indeterminacy introduced by the root system automorphisms introduces the interesting question when the path crosses from one fundamental domain into another. We present an algebraic approach for constructing an indicator function containing this information.