Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

In the Coxeter group W® generated by the root system R, let T® be the number of conjugacy classes having no eigenvalue 1 and let S® be the number of conjugacy classes having no eigenvalue -1. The algebra H{R) of observables of the rational Calogero model based on the root system R possesses T® independent traces, the same algebra considered as an associative superalgebra with respect to a certain natural parity possesses S® even independent supertraces and no odd trace or supertrace. The numbers T® and S® are determined for all irreducible root systems (hence for all root systems). It is shown that T® =< S®, and T® = S® if and only if superalgebra H® contains a Klein operator (or, equivalently, W® containes -1).