Zeta Functions of Root Systems

Abstract

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 A r , 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.

On multiple higher Mahler measures and Witten zeta values associated with semisimple Lie algebras

“…Some explicit formulas for ζ W (2k, g) (k ∈ N) were given by Mordell [9], Zagier [16], Subbarao and Sitaramachandrarao [12] and Gunnells and Sczech [1]. Further Matsumoto "CNTP-6-4-A2-SASAKI" -2013/6/3 -12:58 -page 773 -#3 and Tsumura [8] and Komori et al [3] introduced the multi-variable Witten zeta-functions associated with semisimple Lie algebras, and evaluated special values at positive integers of those functions, including ζ W (2k; g), for some g explicitly (see [2,4,5,8]). …”

On Witten Multiple Zeta-Functions Associated With Semi-Simple Lie Algebras Iv

“…Here we use the same method as introduced in our previous papers [5,7] by making use of polylogarithms as follows. Replacing x by −xe iθ and moving the terms corresponding to l + 3m = 0 of the first member on the left-hand side of the above equation to the right-hand side, we have…”

“…In [5,9] we defined the multi-variable version of for s = (s α ) α∈ + ∈ ‫ރ‬ n , where n is the number of all positive roots. In the case that g is of type X r , we call (1.1) the zeta-function of the root system of type X r , and also denote it by ζ r (s; X r ), where Witten's motivation of introducing the above zeta-functions is to express the volumes of certain moduli spaces in terms of special values of ζ W (s; g).…”

“…In our previous work [5,10], we defined the Bernoulli polynomials of root systems and proved a formula which expresses C W (2k, g) in terms of those Bernoulli polynomials. Consequently, we were able to obtain a certain generalisation of (1.4).…”

“…We further gave some functional relations for zeta-functions of root systems, which include (1.4) as special value-relations. In fact, we studied explicit functional relations for zeta-functions of A r type in [16,5], and of B r and C r types in [10,7] (see also [6]). …”