ZETA-FUNCTIONS OF ROOT SYSTEMS AND POINCARE POLYNOMIALS OF WEYL GROUPS

We consider a certain linear combination of zeta-functions of root systems for a root system. Showing two different expressions of this linear combination, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a generalization of the authors' previous result (Proc. London Math. Soc. 100 (2010), 303-347). We present several explicit examples of such functional relations. We give a criterion of the non-vanishing of the signed sum, in terms of Poincare polynomials of associated Weyl groups. Moreover we prove a certain converse theorem.