Zeta-functions of root systems

Abstract. In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1, 2, 3) which include what is called Witten’s volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups. 1