A survey on the theory of multiple Bernoulli polynomials and multiple $L$-functions of root systems (Infinite Analysis 2010 Developments in Quantum Integrable Systems)

TSUMURAgive the volumes of certain moduli spaces of flat connections, and physically, the 0-th orders of the partition functions of two dimensional quantum gauge theories.Assume that s is an even positive integer 2k .Witten and Zagier showed that their values are in \mat hbb{ Q} $\pi $^{ | \t r i angl e_{ +} | 2k} , where \t r i angl e_{ +} denotes the set of all positive roots.Euler already evaluated them in the A_{1} case.The A_{2} case was first studied by Tornheim [33] and Mordell [29] independently, and further considered by several authors [7, 31, 34].In [32], Szenes gave a certain algorithm for the computation in general cases, from the viewpoint of hyperplane arrangements.Gunnells and Sczech also gave another general algorithm and the explicit forms in the A_{3} case as an application [6].This article is a survey on a new approach to this problem proposed in [11, 12, 15-18, 22, 28] and is an extended and updated version of the informal articles [13, 14].We will introduce generalizations of Bernoulli polynomials and zeta-functions associated with root systems, which include the Riemann zeta-function, the Euler-Zagier zeta-functions and the Witten zeta-functions.Furthermore we will develop a theory similar to that of