ON q-ANALOGUES OF ZETA FUNCTIONS OF ROOT SYSTEMS

Abstract

Komori, Matsumoto and Tsumura introduced a zeta function ζ_r (s, Δ) associated with a root system Δ. In this paper, we introduce a q-analogue of this zeta function, denoted by ζ_r (s, a, Δ; q), and investigate its properties. We show that a ‘Weyl group symmetric' linear combination of ζ_r (s, a, Δ; q) can be written as a multiple integral over a torus involving functions Ψ_s. For positive integers k, functions Ψ_k 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 Ψ_k, which are q-analogues of explicit expressions of Witten's volume formula. We also introduce a two-parameter deformation of the zeta function ζ_r (s, Δ) and study its properties.