ON <i>q</i>-ANALOGUES OF ZETA FUNCTIONS OF ROOT SYSTEMS

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  • KATO Masaki
    Department of General Education National Institute of Technology Toyama College

抄録

<p>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.</p>

収録刊行物

  • 九州数学雑誌

    九州数学雑誌 76 (2), 451-475, 2022

    九州大学大学院数理学研究院

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