Spherical functions for small K-types

2017/10/09 by Hiroshi Oda, Oda, Hiroshi, Nobukazu Shimeno +1
Mathematics · #22E45 #33C67 #43A90 #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1710.02975

openalex publication_date 2017/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (π,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×_K V is obtained from Opdam's theory on hypergeometric Fourier transforms.

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