%0 Journal Article
%T On Witten multiple zeta-functions associated with semisimple Lie algebras IV
%A Yasushi Komori
%A Kohji Matsumoto
%A Hirofumi Tsumura
%J Mathematics
%D 2009
%I arXiv
%X In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.
%U http://arxiv.org/abs/0907.0972v1