Hypergeometric functions of type $BC$ and standard multiplicities

TL;DR

This paper investigates hypergeometric functions of type BC with standard multiplicities, establishing positivity, estimates, and characterizations, and extends properties of spherical functions on symmetric spaces to vector bundles.

Contribution

It introduces new positivity and estimate results for BC-type hypergeometric functions with non-positive multiplicities, extending classical spherical function properties.

Findings

Positivity properties of hypergeometric functions of type BC.

Sharp estimates for these functions.

Extension of spherical function properties to vector bundles.

Abstract

We study the Heckman-Opdam hypergeometric functions associated to a root system of type $BC$ and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates which imply a characterization of the bounded hypergeometric functions. As an application, our results extend known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type $G/K$ to spherical functions over homogeneous vector bundles on $G/K$ which are associated to certain small $K-$types.