Spherical functions for small $K$-types

TL;DR

This paper classifies small $K$-types in semisimple Lie groups and expresses their spherical functions using hyperbolic cosines and hypergeometric functions, enabling an inversion formula for the spherical transform.

Contribution

It provides a complete classification of small $K$-types and explicit formulas for their spherical functions, extending Opdam's hypergeometric Fourier transform theory.

Findings

All small $K$-types classified for simple Lie groups.

Spherical functions expressed as products of hyperbolic cosines and hypergeometric functions.

Derived inversion formula for the spherical transform.

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.