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

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 $χ$-spherical functions on Hermitian symmetric spaces $U/K$ where $χ$ 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 $χ$-spherical functions. The motive comes from the Paley-Wiener type theorem on line bundles over Hermitian symmetric spaces.

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