M-series and Kloosterman–Selberg zeta functions for ℝ-rank one groups

Abstract

For an arbitrary Lie group G of real rank one, we give a formula for the Fourier coefficient Dχχ′(ξ,υ) of the M-series (a type of Poincaré series) defined in [17], in terms of Kloosterman–Selberg zeta functions ξχ,χ′,ξ(μ). As a consequence we show that the meromorphic continuation of ξχ,χ′,ξ(υ) to ℂ follows from the meromorphic continuation of the M-series. We also give a description of the pole set in the region Re υ ≥ 0.