Spherical functions for small $K$-types

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.

43 pages

References in corpus (1)

• Representations with Small $K$ Types