Hypergeometric functions of type $BC$ and standard multiplicities

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.

The present paper subsumes and extends the results of the preprint arXiv:1705.00277. With respect to arXiv:2008.00337: Corrected Proposition 3.5. Some computations added in Appendix B

References in corpus (2)

• Differential-difference operators and radial part formulas for non-invariant elements
• Inversion formula for the hypergeometric Fourier transform associated with a root system of type $BC$