An application of hypergeometric shift operators to the chi-spherical Fourier transform

TL;DR

This paper explores how hypergeometric shift operators can transform hypergeometric functions related to Hermitian symmetric spaces, enabling the derivation of exponential estimates crucial for Paley-Wiener theorems.

Contribution

It introduces a method to shift negative multiplicities to positive ones for hypergeometric functions, facilitating the application of existing results.

Findings

Established exponential estimates for $hi$-spherical functions.

Connected shift operators to the Paley-Wiener theorem.

Enhanced understanding of hypergeometric functions on root systems.

Abstract

We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the $BC_n$ type root system and some negative multiplicities. Those hypergeometric functions are connected to the $\chi$-spherical functions on Hermitian symmetric spaces $U/K$ where $\chi$ is a nontrivial character of $K$. We apply shift operators to the hypergeometric functions to move negative multiplicities to positive ones. This allows us to use many well-known results of the hypergeometric functions associated with positive multiplicities. In particular, we use this technique to achieve exponential estimates for the $\chi$-spherical functions. The motive comes from the Paley-Wiener type theorem on line bundles over Hermitian symmetric spaces.