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Spherical Functions for Small K-Types

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 290)

Abstract

For a connected semisimple real Lie group G of non-compact type, Wallach introduced a class of K-types called small. We classify all small K-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small K-type \((\pi ,V)\) can be expressed as a product of hyperbolic cosines and a Heckman–Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on \(G\times _K V\) is obtained from Opdam’s theory on hypergeometric Fourier transforms.

Keywords

Small K-types Spherical functions Hypergeometric functions 

1991 Mathematics Subject Classification

22E45 33C67 43A90 

Notes

Acknowledgements

The authors wish to thank Professor Hiroyuki Ochiai for helpful comments on an earlier version of this paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of EngineeringTakushoku UniversityHachiojiJapan
  2. 2.School of Science and TechnologyKwansei Gakuin UniversitySandaJapan

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