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

  • Roberto J. MiatelloEmail author
  • Nolan R. Wallach
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

For an arbitrary Lie group G of real rank one, we give a formula for the Fourier coefficient \(D_{\chi ^{{\prime}}}^{\chi }(\xi,\nu )\) of the M-series (a type of Poincaré series) defined in [17], in terms of Kloosterman–Selberg zeta functions \(\zeta _{\chi,\chi ^{{\prime}},\xi }(\mu )\). As a consequence we show that the meromorphic continuation of \(\zeta _{\chi,\chi ^{{\prime}},\xi }(\nu )\) to \(\mathbb{C}\) follows from the meromorphic continuation of the M-series. We also give a description of the pole set in the region \(\mathop{Re}\nolimits \nu \geq 0\).

Keywords

Kloosterman sum Kloosterman–Selberg zeta function Fourier coefficient Whittaker vector rank one Lie group 

Mathematics Subject Classification

11F30 11F70 11L05 11M36 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.FaMAF-CIEM, Facultad de Matemática, Astronomia y FisicaUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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