2022 年 76 巻 2 号 p. 451-475
Komori, Matsumoto and Tsumura introduced a zeta function ζr (s, Δ) associated with a root system Δ. In this paper, we introduce a q-analogue of this zeta function, denoted by ζr (s, a, Δ; q), and investigate its properties. We show that a ‘Weyl group symmetric' linear combination of ζr (s, a, Δ; q) can be written as a multiple integral over a torus involving functions Ψs. For positive integers k, functions Ψk can be regarded as q-analogues of the periodic Bernoulli polynomials. When Δ is of type A2 or A3, the linear combinations can be expressed as the functions Ψk, which are q-analogues of explicit expressions of Witten's volume formula. We also introduce a two-parameter deformation of the zeta function ζr (s, Δ) and study its properties.