Analysis on the Crown Domain


This paper is a further development of complex methods in harmonic analysis on semi-simple Lie groups [AG], [BeR], [KrS1,2]. We study the growth behaviour of the holomorphic extension of the orbit map of the spherical vector of an irreducible spherical representation of a real reductive group G when approaching the boundary of the crown domain of the Riemannian symmetric space G/K. As an application, we prove that Maaß cusp forms have exponential decay.

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Correspondence to Bernhard Krötz.

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During the preparation of this paper the second named author was partially supported by a Pionier Grant of the Netherlands Organization for Scientific Research (NWO). Part of this research was carried out in the fall of 2004, during which period both authors enjoyed the hospitality of the Research Institute for the Mathematical Sciences in Kyoto, Japan. It is our pleasure to thank the RIMS for its hospitality and for the stimulating environment it offers.

Received: August 2006, Revision: June 2007, Accepted: June 2007

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Krötz, B., Opdam, E. Analysis on the Crown Domain. GAFA Geom. funct. anal. 18, 1326–1421 (2008).

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Keywords and phrases:

  • Crown domain
  • harmonic analysis
  • Maaß cusp forms
  • hypergeometric functions associated to root systems

AMS Mathematics Subject Classification:

  • 22Exx
  • 11xx